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Infinite Limits and Continuity

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Given the function f(x)={arctan⁡x+π/2if x≤02xif 0<x≤e1ln⁡xif x>ef(x)=\begin{cases} \arctan x+\pi/2&\text{if }x \le 0\\ \frac{2}{x}&\text{if }0<x\le e\\ \frac{1}{\ln x}&\text{if }x>e \end{cases}, which of the following statements is true about f(x)f(x)?
More Limits at Infinity Questions:
Limits at Infinity
lim⁡x→∞x2+2x2−3x+1\displaystyle\lim_{x\to\infty}\dfrac{x^2+2}{x^2-3x+1}
Limits
State the asymptotes of the function f(x)=3x2−54x2−1f\left(x\right)=\frac{3x^2-5}{4x^2-1}.
Limit as x→±∞
What is the value of the limit lim⁡x→−∞x2+2−5x3x−1\lim\limits_{x\rightarrow -\infty}\frac{\sqrt{x^2+2}-5x}{3x-1}?
Limits at Infinity
lim⁡x→∞x2+2x2−3x+1\displaystyle\lim_{x\to\infty}\dfrac{x^2+2}{x^2-3x+1}
Limits at Infinity
lim⁡x→−∞2−x−2x23x2+5\displaystyle\lim_{x\rightarrow -\infty}\dfrac{2-x-2x^2}{3x^2+5}
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Limits at Infinity
lim⁡x→−∞2−x−2x23x2+5\displaystyle\lim_{x\rightarrow -\infty}\dfrac{2-x-2x^2}{3x^2+5}
Limits at Infinity
lim⁡x→∞x2+2x2−3x+1\displaystyle\lim_{x\to\infty}\dfrac{x^2+2}{x^2-3x+1}
Indeterminate Limits
lim⁡x→∞ 24x6−x4+2x3−13x6+2x5+3=\displaystyle \lim_{x\to\infty}\ \frac{24x^6-x^4+2x^3-1}{3x^6+2x^5+3}=
Limit as x→±∞
What is the value of the limit lim⁡x→−∞x2+2−5x3x−1\lim\limits_{x\rightarrow -\infty}\frac{\sqrt{x^2+2}-5x}{3x-1}?
What is lim⁡x→∞f(x)\displaystyle\lim_{x\rightarrow\infty}f(x) if lim⁡x→∞g(x)=8\displaystyle\lim_{x\rightarrow \infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)-2}=4
Limits at Infinity: Limits with Given Values
Find lim⁡x→∞f(x)\displaystyle\lim_{x\rightarrow\infty}f(x) if lim⁡x→∞g(x)=8\displaystyle\lim_{x\rightarrow\infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)-2}=4.
Evaluate the following limit
lim⁡x→−∞2−x−2x23x2+5\displaystyle \lim_{x\rightarrow -\infty}\frac{2-x-2x^2}{3x^2+5}
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Limits: Asymptotes
Find the horizontal and vertical asymptotes of y=tan⁡(4x)y=\tan(4x).
Evaluating Limits
Evaluate the following limit, or state that it does not exist and why.
lim⁡x→∞4x6−7x3+13x3+2x−2\lim_{x \rightarrow \infty} \frac{\sqrt{4x^6 - 7x^3 + 1}}{3x^3 + 2x - 2}
∞ - ∞ Limits with Square Root
lim⁡x→∞ x2+2x−x2−3x=\displaystyle\lim_{x\to\infty}\ \sqrt{x^2+2x}-\sqrt{x^2-3x}=
Limits at Infinity: Limits with Given Values
Find lim⁡x→∞f(x)\displaystyle\lim_{x\rightarrow\infty}f(x) if lim⁡x→∞g(x)=8\displaystyle\lim_{x\rightarrow\infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)-2}=4.
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Limits at Infinity
Evaluate lim⁡ x→∞−45−x\lim\ _{x\rightarrow\infty}\frac{-4}{5-x}
Limits at Infinity: Evaluating limits
Determine lim⁡ x→−∞4x2x+2\lim\ _{x\rightarrow-\infty}\frac{\sqrt{4x^2}}{x+2}
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Limit as x→±∞
What is the value of the limit lim⁡x→−∞x2+2−5x3x−1\lim\limits_{x\rightarrow -\infty}\frac{\sqrt{x^2+2}-5x}{3x-1}?
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Limit as x→±∞
What is the value of the limit lim⁡x→−∞x2+2−5x3x−1\lim\limits_{x\rightarrow -\infty}\frac{\sqrt{x^2+2}-5x}{3x-1}?
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Limits: Asymptotes
Find the horizontal and vertical asymptotes of y=tan⁡(4x)y=\tan(4x).
Evaluating Limits
Evaluate the following limit, or state that it does not exist and why.
lim⁡x→∞4x6−7x3+13x3+2x−2\lim_{x \rightarrow \infty} \frac{\sqrt{4x^6 - 7x^3 + 1}}{3x^3 + 2x - 2}
∞ - ∞ Limits with Square Root
lim⁡x→∞ x2+2x−x2−3x=\displaystyle\lim_{x\to\infty}\ \sqrt{x^2+2x}-\sqrt{x^2-3x}=
Limits at Infinity: Limits with Given Values
Find lim⁡x→∞f(x)\displaystyle\lim_{x\rightarrow\infty}f(x) if lim⁡x→∞g(x)=8\displaystyle\lim_{x\rightarrow\infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)-2}=4.
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Limits at Infinity
Evaluate lim⁡ x→∞−45−x\lim\ _{x\rightarrow\infty}\frac{-4}{5-x}
Limits at Infinity: Evaluating limits
Determine lim⁡ x→−∞4x2x+2\lim\ _{x\rightarrow-\infty}\frac{\sqrt{4x^2}}{x+2}
Limits at Infinity: Evaluating limits
Practice: Evaluating limits
Determine lim⁡ x→−∞4x2x+2\lim\ _{x\rightarrow-\infty}\frac{\sqrt{4x^2}}{x+2}
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Evaluating Limits
Evaluate the following limit, or state that it does not exist and why.
lim⁡x→∞4x6−7x3+13x3+2x−2\lim_{x \rightarrow \infty} \frac{\sqrt{4x^6 - 7x^3 + 1}}{3x^3 + 2x - 2}
Horizontal Asymptote
Practice: Horizontal Asymptote
Find the horizontal asymptote(s), if any, of y=16 x7+x3−2x2+12x7−x5+x3+4x−5y=\frac{16\ x^7+x^3-2x^2+1}{2x^7-x^5+x^3+4x-5}
Limits at Infinity: Evaluating limits
Practice: Evaluating limits
Determine lim⁡ x→−∞4x2x+2\lim\ _{x\rightarrow-\infty}\frac{\sqrt{4x^2}}{x+2}
Limits at Infinity
Evaluate lim⁡ x→∞−45−x\lim\ _{x\rightarrow\infty}\frac{-4}{5-x}
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Practice: All Asymptotes
Q.\textbf{Q.} Find all the vertical, horizontal and slant asymptotes of f(x)=x3+x−2x−x2+1\displaystyle f(x)=\frac{x^3+x-2}{x-x^2}+1.
Limits at Infinity
Find all asymptotes of f(x)=3x2+1\displaystyle f(x)=\frac{3}{x^{2}+1}.
Limits at Infinity
Find the horizontal asymptote of f(x)=4x2+1−2x\displaystyle f(x)=\sqrt{4x^2+1}-2x.
Q.\textbf{Q.} Evaluate the limit lim⁡s→∞s3+12s3+3s−1\displaystyle\lim_{s\rightarrow \infty}\frac{s^{3}+1}{2s^{3}+3s-1}
Find all asymptotes for f(x)=3x2+x+1x2−1\displaystyle f(x)=\frac{3x^2+x+1}{x^2-1} . Enter as comma-separated values (no spaces in between), smaller value first
The limit lim⁡x→∞x2+2−5x3x−1\displaystyle\lim_{x\to\infty}\frac{\sqrt{x^2+2}-5x}{3x-1} is equal to:
Evaluate the following limit: lim⁡x→+∞2x4−x3+10x2x2−3x4\displaystyle\lim_{x\rightarrow+\infty}\frac{2x^4-x^3+10x^2}{x^2-3x^4}
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Q. \textbf{Q.} Find aa and bb such that lim⁡x→∞4x2+2x+9ax3+bx2+1=2\displaystyle\lim_{x\rightarrow\infty}\frac{4x^2+2x+9}{ax^3+bx^2+1}=2.
Limit at Infinity with Roots
Evaluate a) lim⁡x→∞5x2−4−3x1−9x\displaystyle\lim_{x\to\infty}\frac{\sqrt{5x^2-4}-3x}{1-9x} and b) lim⁡x→−∞5x2−4−3x1−9x\displaystyle\lim_{x\to-\infty}\frac{\sqrt{5x^2-4}-3x}{1-9x}.
Limit at Infinity with Polynomials
Evaluate lim⁡x→∞x3+12x3−x2+x+4\displaystyle \lim_{x\rightarrow\infty}\frac{x^3+1}{2x^3-x^2+x+4}
Limits at Infinity: Limits with Given Values
Find lim⁡x→∞f(x)\displaystyle\lim_{x\rightarrow\infty}f(x) if lim⁡x→∞g(x)=8\displaystyle\lim_{x\rightarrow\infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)-2}=4.
Limits at Infinity: Domain and Range
(Original question is written in long answer. Multiple choice has been used here for convenience)
Let f(x)=x2−6x−3\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}.
State all values of aa, bb, and cc such that 4x2+2x+9ax3+bx2−c\dfrac{4x^2+2x+9}{ax^3+bx^2-c} has the following properties:
A single horizontal asymptote at y=2y=2
A vertical asymptote at x=2x=2 and a vertical asymptote at x=−2x=-2
Limits at Infinity
lim⁡x→∞x2+2x2−3x+1\displaystyle\lim_{x\to\infty}\dfrac{x^2+2}{x^2-3x+1}
∞ - ∞ Limits with Square Root
Practice: ∞ - ∞ Limits with Square Root
lim⁡x→∞ x2+2x−x2−3x=\displaystyle\lim_{x\to\infty}\ \sqrt{x^2+2x}-\sqrt{x^2-3x}=
Limits at Infinity
Find the horizontal asymptote(s), if any, of y=16x7+x3−2x2+12x7−x5+x3+4x−5y=\frac{16x^7+x^3-2x^2+1}{2x^7-x^5+x^3+4x-5}.
Limits at Infinity
lim⁡x→−∞x2+10x−8x+56x−4\displaystyle\lim_{x\rightarrow-\infty}\frac{\sqrt{x^2+10x}-8x+5}{6x-4}
Limits at Infinity
lim⁡x→+∞x2+10x−8x+24x−2\displaystyle\lim_{x\rightarrow+\infty}\frac{\sqrt{x^2+10x}-8x+2}{4x-2}
lim⁡x→−∞2−x−2x23x2+5\displaystyle\lim_{x\rightarrow -\infty}\dfrac{2-x-2x^2}{3x^2+5}
lim⁡x→∞x2+2x2−3x+1\displaystyle\lim_{x\to\infty}\dfrac{x^2+2}{x^2-3x+1}
lim⁡x→−∞2−x−2x23x2+5\displaystyle\lim_{x\rightarrow -\infty}\dfrac{2-x-2x^2}{3x^2+5}
Evaluate lim⁡x→−∞2x+100625x2+3x−12\displaystyle\lim_{x\to-\infty} \frac{2x+100}{\sqrt{625x^2 +3x -12}}
Evaluate lim⁡x→−∞2x−8025x2+x−5\displaystyle\lim_{x\to-\infty} \frac{2x-80}{\sqrt{25x^2 +x -5}}
Limits at Infinity
Find the asymptote(s), if any, of
y=x3−1x2y=\dfrac{x^3 -1}{x^2}
Limits at Infinity
Find a and b such that
lim⁡x→∞4x2+2x+9ax3+bx2+1=2\lim\limits_{x\to\infty}\frac{4x^2+2x+9}{ax^3+bx^2+1}=2
Indeterminate Limits
Find lim⁡x→∞f(x)\lim\limits_{x\to \infty} f(x) if lim⁡x→∞g(x)=8\lim\limits_{x\to\infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\lim\limits_{x\to\infty}\frac{g(x)}{f(x)-2}=4.
Indeterminate Limits
Evaluate the limit:
lim⁡x→∞5x2+83x2+2−6x3\lim\limits_{x \to \infty}\frac{5x^2+8}{3x^2+2-6x^3}
Practice: Extracting Highest Power Indeterminate Forms
For limits at infinity including square roots only multiply by conjugate if infinities are cancelling out by subtraction. Otherwise, extract the term with the greatest exponent: Evaluate lim⁡x→∞x2−10x−x2.\lim_{x\to\infty}\limits\sqrt{x^2 - 10x} - x^2.
Practice: Dividing by Highest Power Indeterminate Forms
Dividing Numerator and Denominator by Highest Power: Evaluate lim⁡x→∞2x2−13x2+x+5\lim_{x\to\infty}\limits\dfrac{2x^2 - 1}{3x^2 + x + 5}
Limit with square root
Evaluate lim⁡x→−∞3x6−x+12x3\displaystyle\lim_{x\to-\infty}\frac{\sqrt{3x^6-x+1}}{2x^3}.
*Be careful with the negative sign in −∞-\infty! Check out the hint for more info.
Determine the value of lim⁡x→∞8x9+3x2−133x3−x\displaystyle \lim_{x\rightarrow\infty}\frac{\sqrt[3]{8x^9+3x^2-1}}{3x^3-x}. Express your answer as a fraction in lowest terms.
What is the value of the limit lim⁡x→−∞x2+2−5x3x+1\lim_{x\to-\infty}\frac{\sqrt{x^2+2}-5x}{3x+1} ?
Evaluate lim⁡x→−∞2x+100625x2+3x−12\displaystyle\lim_{x\to-\infty} \frac{2x+100}{\sqrt{625x^2 +3x -12}}
Limit at Infinity with Polynomials
Evaluate lim⁡x→∞x3+12x3−x2+x+4\displaystyle \lim_{x\rightarrow\infty}\frac{x^3+1}{2x^3-x^2+x+4}
Limits at Infinity
Find all asymptotes of f(x)=3x2+1\displaystyle f(x)=\frac{3}{x^{2}+1}.
Practice: All Asymptotes
Q.\textbf{Q.} Find all the vertical, horizontal and slant asymptotes of f(x)=x3+x−2x−x2+1\displaystyle f(x)=\frac{x^3+x-2}{x-x^2}+1.
Limits at Infinity
Find the horizontal asymptote of f(x)=4x2+1−2x\displaystyle f(x)=\sqrt{4x^2+1}-2x.
Evaluate the following limit: lim⁡x→+∞2x4−x3+10x2x2−3x4\displaystyle\lim_{x\rightarrow+\infty}\frac{2x^4-x^3+10x^2}{x^2-3x^4}
The limit lim⁡x→∞x2+2−5x3x−1\displaystyle\lim_{x\to\infty}\frac{\sqrt{x^2+2}-5x}{3x-1} is equal to:
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Q. \textbf{Q.} Find aa and bb such that lim⁡x→∞4x2+2x+9ax3+bx2+1=2\displaystyle\lim_{x\rightarrow\infty}\frac{4x^2+2x+9}{ax^3+bx^2+1}=2.
Limit at Infinity with Roots
Evaluate a) lim⁡x→∞5x2−4−3x1−9x\displaystyle\lim_{x\to\infty}\frac{\sqrt{5x^2-4}-3x}{1-9x} and b) lim⁡x→−∞5x2−4−3x1−9x\displaystyle\lim_{x\to-\infty}\frac{\sqrt{5x^2-4}-3x}{1-9x}.
Limits at Infinity: Limits with Given Values
Find lim⁡x→∞f(x)\displaystyle\lim_{x\rightarrow\infty}f(x) if lim⁡x→∞g(x)=8\displaystyle\lim_{x\rightarrow\infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)-2}=4.
Q.\textbf{Q.} Evaluate the limit lim⁡s→∞s3+12s3+3s−1\displaystyle\lim_{s\rightarrow \infty}\frac{s^{3}+1}{2s^{3}+3s-1}
Find all asymptotes for f(x)=3x2+x+1x2−1\displaystyle f(x)=\frac{3x^2+x+1}{x^2-1} . Enter as comma-separated values (no spaces in between), smaller value first
What is the value of the limit lim⁡x→−∞x2+2−5x3x+1\lim_{x\to-\infty}\frac{\sqrt{x^2+2}-5x}{3x+1} ?
Find the horizontal asymptote(s), if any, of y=16 x7+x3−2x2+12x7−x5+x3+4x−5y=\frac{16\ x^7+x^3-2x^2+1}{2x^7-x^5+x^3+4x-5}. Enter your answer as an equation of a line, e.g. y=−6y=-6.
Evaluate the limit, lim⁡x→+∞x4−x3+45x2x2−2x4\displaystyle\lim_{x\rightarrow+\infty}\frac{x^4-x^3+45x^2}{x^2-2x^4}.
Limit as x→±∞
What is the value of the limit lim⁡x→−∞x2+2−5x3x−1\lim\limits_{x\rightarrow -\infty}\frac{\sqrt{x^2+2}-5x}{3x-1}?
Limits at Infinity
Determine the value of the limit (simplify)
lim⁡x→−∞arctan⁡(1−x−x32x2−9) \displaystyle \lim_{x \to -\infty} \arctan \left( \frac{1-x-x^3}{2x^2-9} \right)
Limits at Infinity
What is the value of the limit lim⁡x→−∞x2+2+5x2x+1?\lim\limits_{x\rightarrow-\infin}\frac{\sqrt{x^2+2}+5x}{2x+1}?
Evaluate the following limit.
lim⁡x→∞5x2+83x2+2−6x3\displaystyle \lim_{x\rightarrow \infty} \frac{5x^2+8}{3x^2+2-6x^3}
Determine lim⁡ x→−∞4x2x+2\lim\ _{x\rightarrow-\infty}\frac{\sqrt{4x^2}}{x+2}.
What is lim⁡x→∞f(x)\displaystyle \lim_{x\rightarrow \infty}f(x) if lim⁡x→∞g(x)=8\displaystyle \lim_{x\rightarrow \infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle \lim_{x\rightarrow \infty}\frac{g(x)}{f(x)-2}=4?
Determine the value of lim⁡x→∞4x3+πx5x3+6x2−100\displaystyle \lim_{x\rightarrow \infty} \frac{4x^3+\pi x}{5x^3+6x^2-100}. Express your answer as a fraction in lowest terms.
Determine the value of lim⁡x→−∞6x−75x2−2x+9\displaystyle \lim_{x \to -\infty} \frac{6x-7}{\sqrt{5x^2-2x+9}}. Express your answer as a fraction in lowest terms. If your answer is a negative number, put the negative sign in front of the entire fraction.
What is the value of lim⁡x→∞x2+2x2−3x+1\displaystyle \lim_{x\rightarrow \infty}\frac{x^2+2}{x^2-3x+1}?
Evaluate the limit, lim⁡x→+∞2x4−x3+10x2x2−3x4\displaystyle\lim_{x\rightarrow+\infty}\frac{2x^4-x^3+10x^2}{x^2-3x^4}.
🌶️ TOUGH! True or false, in the case of a horizontal asymptote, if lim⁡x→∞f(x)=2\lim_{x\rightarrow\infty}f(x)=2 then f(x)f(x) never intersects y=2y=2.
State the vertical and horizontal asymptotes of the function f(x)=3x2−54x2−1f(x)=\frac{3x^2-5}{4x^2-1}.
Vertical asymptote x= ±1/2. Horizontal asymptote f(x)=3/4
Find all the vertical and horizontal asymptotes for f(x)=3x2+x+1x2−1f(x)=\frac{3x^2+x+1}{x^2-1} .
Find all vertical and horizontal asymptotes for f(x)=3x+1x−1f(x)=\frac{3x+1}{x-1} . Enter the asymptotes in the form of the equation of a line, e.g. y=−6y=-6.
What is the value of lim⁡x→−∞2−x−2x23x2+5\displaystyle \lim_{x\rightarrow -\infty}\frac{2-x-2x^2}{3x^2+5}?
🦊 TRICKY! Evaluate the limit, lim⁡x→∞x2−10x−x2\displaystyle\lim_{x\rightarrow \infty} \sqrt{x^2-10x}-x^2.
Evaluate the limit, lim⁡x→∞2x2−13x2+x+5\displaystyle\lim_{x\rightarrow\infty}\frac{2x^2-1}{3x^2+x+5}.
Practice: Limit as x→±∞
Evaluate lim⁡x→−∞2x−14x2+x+x+5\lim\limits_{x\rightarrow -\infty}\frac{2x-1}{\sqrt{4x^2+x}+x+5}
Limits: Asymptotes
Find the horizontal and vertical asymptotes of y=tan⁡(4x)y=\tan(4x).
Limits: Asymptotes
Find the vertical and horizontal asymptotes of y=ln⁡(tan⁡−1x)y=\ln(\tan^{-1}x).
Find aa and bb such that lim⁡x→∞4x2+2x+9ax3+bx2+1=2\displaystyle \lim_{x\rightarrow\infty}\frac{4x^2+2x+9}{ax^3+bx^2+1}=2
Limits
State the asymptotes of the function f(x)=3x2−54x2−1f\left(x\right)=\frac{3x^2-5}{4x^2-1}.
Limits at Infinity: Evaluating limits
Practice: Evaluating limits
Determine lim⁡ x→−∞4x2x+2\lim\ _{x\rightarrow-\infty}\frac{\sqrt{4x^2}}{x+2}
Horizontal Asymptote
Practice: Horizontal Asymptote
Find the horizontal asymptote(s), if any, of y=16 x7+x3−2x2+12x7−x5+x3+4x−5y=\frac{16\ x^7+x^3-2x^2+1}{2x^7-x^5+x^3+4x-5}
lim⁡x→∞x2+2x−x2+1\displaystyle\lim_{x\rightarrow\infty}\sqrt{x^2+2x}-\sqrt{x^2+1}
Evaluate the following limit
lim⁡x→−∞2−x−2x23x2+5\displaystyle \lim_{x\rightarrow -\infty}\frac{2-x-2x^2}{3x^2+5}
Evaluate the following limit
lim⁡x→∞x2x2+x+1\displaystyle \lim_{x\rightarrow \infty}\frac{x}{\sqrt{2x^2+x+1}}
Limits at Infinity
Evaluate lim⁡ x→∞−45−x\lim\ _{x\rightarrow\infty}\frac{-4}{5-x}
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Practice: $\frac{\infty}{\infty}$ Limit with Square Root
Practice: ∞/∞ Limit with Square Root
Evaluate lim⁡x→−∞3x6−x+12x3\displaystyle\lim_{x\to-\infty}\frac{\sqrt{3x^6-x+1}}{2x^3}.
*Be careful with the negative sign in −∞-\infty! Check out the hint for more info.
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Indeterminate Limits
Practice: ∞/∞ Limit
lim⁡x→∞ 24x6−x4+2x3−13x6+2x5+3=\displaystyle \lim_{x\to\infty}\ \frac{24x^6-x^4+2x^3-1}{3x^6+2x^5+3}=
Each question is worth 3 marks.
Evaluating Limits
Evaluate the following limit, or state that it does not exist and why.
lim⁡x→∞4x6−7x3+13x3+2x−2\lim_{x \rightarrow \infty} \frac{\sqrt{4x^6 - 7x^3 + 1}}{3x^3 + 2x - 2}
Limits and Continuity
Evaluate the following limits:
a) lim⁡x→4x−4x−4\displaystyle \lim_{x \rightarrow 4} \frac{\sqrt{x} - \sqrt{4}}{x - 4}
b) lim⁡x→∞x2−5x+134x4+3x−21\displaystyle \lim_{x \rightarrow \infty} \frac{x^2 - 5x + 13}{\sqrt{4x^4 +3x - 21}}
Limits at Infinity
Find the slant asymptote to the curve y=x3+x2+1x2+1\displaystyle y=\frac{x^3+x^2+1}{x^2+1}
Find the slant asymptote to the curve y=x3−12x2+16x−17x2−10x+25\displaystyle y=\frac{x^3-12x^2+16x-17}{x^2-10x+25}
lim⁡x→∞x2+2x2−3x+1\displaystyle\lim_{x\to\infty}\dfrac{x^2+2}{x^2-3x+1}
Limits at Infinity
lim⁡x→−∞2−x−2x23x2+5\displaystyle\lim_{x\rightarrow -\infty}\dfrac{2-x-2x^2}{3x^2+5}
What is lim⁡x→∞f(x)\displaystyle\lim_{x\rightarrow\infty}f(x) if lim⁡x→∞g(x)=8\displaystyle\lim_{x\rightarrow \infty}g(x)=8 and lim⁡x→∞g(x)f(x)−2=4\displaystyle\lim_{x\rightarrow\infty}\frac{g(x)}{f(x)-2}=4
lim⁡x→−∞2−x−2x23x2+5\displaystyle\lim_{x\rightarrow -\infty}\dfrac{2-x-2x^2}{3x^2+5}
More The Limit Definition of a Derivative Questions:
Limit definition of the derivative (first principles)
If f(x)=1x,f(x)=\frac{1}{\sqrt{x}},then find the derivative of the function by first principles.
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative): h(t)=216−th\left(t\right)=2\sqrt{16-t}
Definition of the Derivative
Carefully state what the definition of the derivative of a function is at the point x=ax = a .
Definition of a Derivative
Using the definition of the derivative, find the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x
Definition of a Derivative
Using the definition of the derivative, set up the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x, but do not solve!
Limit Definition of a Derivative
Find a function f(x)f(x)and a number aa such that
f′(a)=lim⁡x→0cos⁡x−1xf^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}
Limit Definition of a Derivative
Given that f′(2)=4f^{\prime}(2)=4, find the following limit:
lim⁡x→2x−2f(x)−f(2) \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
Limit definition of a derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative.
f(x)=1x+3 for x≠−3f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
Definition of a Derivative
Using the definition of the derivative, find the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of f(x)=x2f\left(x\right)=\frac{\sqrt{x}}{2} simplifies to
Limit definition of the derivative (first principles)
If f(x)=1x,f(x)=\frac{1}{\sqrt{x}},then find the derivative of the function by first principles.
Practice: Definition of Derivative
Practice: Definition of Derivative
For which pair of f(x)f\left(x\right) and aa does the equality f(a)=lim⁡x→8 x+1−3x−8\displaystyle f\left(a\right)=\lim_{x\to8}\ \frac{\sqrt{x+1}-3}{x-8} hold?
Limit Definition of Derivatives
Which of the following limits represents a derivative?
i) lim⁡a→0 f(x+a)−f(x)x−a\displaystyle \lim_{a\to0}\ \frac{f\left(x+a\right)-f\left(x\right)}{x-a}
ii) lim⁡x→3 x−3x−3\displaystyle \lim_{x\to3}\ \frac{\sqrt{x-3}}{x-3}
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of f(x)=x2f\left(x\right)=\frac{\sqrt{x}}{2} simplifies to
Limit definition of a derivative
Evaluate the following limit: lim⁡x→052030−(5+x)20302x\displaystyle\lim_{x\rightarrow0} \frac{5^{2030}-(5+x)^{2030}}{2x}
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative): h(t)=216−th\left(t\right)=2\sqrt{16-t}
Q.\textbf{Q.} Consider again the function f(x)=x2+1f(x)=x^2+1. What is the instantaneous rate of change of ff at x=2x=2? Answer using the definition of the derivative.
Q.\textbf{Q.} Show that f(x)=∣x−2∣−1f(x)=\lvert x-2\rvert-1 is not differentiable at x=2x=2.
Limit Definition of a Derivative: Working with Given Values
Given that f′(2)=4f^{\prime}(2)=4, find lim⁡x→2x−2f(x)−f(2)\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}.
Limit Definition of a Derivative
Find a function ff and a number aa such that:
f′(a)=lim⁡x→0cos⁡x−1x{f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}}
Limits: Definition with Fractions
Find the derivative of g(x)=12+x\displaystyle g(x)=\frac{1}{2+x} using the definition.
Limit Definition of a Derivative
Q:\textbf{Q:} Find f′(2)f'(2) if f(x)=(2x+5)12f(x)=(2x+5)^{\frac{1}{2}} using the definition of the derivative.
Limit Definition of a Derivative
(Original question is written in long answer. Multiple choice has been used here for convenience)
Let f(x)=xx−3\displaystyle f(x) = \frac{x}{x - 3}. Compute dfdx\displaystyle \frac{df}{dx} using the definition of the derivative. No marks will be given for the use of derivative rules, but you may use them to check your answer.
Limit Definition of a Derivative
Find a function f(x)f(x)and a number aa such that
f′(a)=lim⁡x→0cos⁡x−1xf^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}
Limit Definition of a Derivative
Find the following derivatives using the definition of derivative.
f(x)=2x3+1f(x)=2x^3+1
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=1x2+3\displaystyle f(x)=\frac{1}{x^2+3}
Find the following using the definition of derivative.
f(x)=x3f(x)=\sqrt[3]{x}
Find f′(x)f^{\prime}(x) using the limit definition of the derivative.
f(x)=1x+3 for x≠−3f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
Limit Definition of a Derivative
Evaluate
ddx(−2x2−2x+3)\dfrac{d}{dx}(-2x^2-2x+3)
using the limit definition of derivative.
Limit Definition of a Derivative
Evaluate ddx(−x2+7x−2)\dfrac{d}{dx}(-x^2+7x-2) using the limit definition of derivative.
Limit Definition of a Derivative
Given that f′(2)=4f'(2)=4, find
lim⁡x→2x−2f(x)−f(2)\lim\limits_{x\to 2} \frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
Limit Definition of a Derivative
If f(x) is differentiable everywhere, find A and B.
f(x)={x2+1x≥0Ax+Bx<0f(x)=\begin{cases}x^2+1 & x \geq 0\\ Ax+B&x<0\end{cases}
Limit Definition of a Derivative
Find a function f and a number a such that
f′(a)=lim⁡x→0cos⁡x−1xf'(a)=\lim\limits_{x\to 0}\frac{\cos x-1}{x}
Derivatives: Limit Definition
True or false: f(x)=∣x−3∣−1f(x) = |x-3|-1 is differentiable at x=3.x=3.
Limit definition of a derivative
Find f′(2)f'(2) if 2f(2)−2f(2+h)h=3h−2\frac{2f(2)-2f(2+h)}{h}=\frac{3}{h-2} for all h near 0.
Practice: Differentiability
Practice: Differentiability
Which of the following is/are NOT differentiable at x=0x=0?
(Select all that apply)
Find the derivative of f(x)=1x+3\displaystyle f(x)=\frac{1}{x+3}, using the limit definition.
Limit Definition of a Derivative
Use the limit definition to find the derivative of g(x)=xg(x)=\sqrt{x}.
Limit Definition of a Derivative
Given that f′(2)=4f^{\prime}(2)=4, find the following limit:
lim⁡x→2x−2f(x)−f(2) \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}
Limit Definition of a Derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=1x2+3\displaystyle f(x)=\frac{1}{x^2+3}
Limit definition of a derivative
Evaluate the following limit: lim⁡x→052030−(5+x)20302x\displaystyle\lim_{x\rightarrow0} \frac{5^{2030}-(5+x)^{2030}}{2x}
Limit Definition of a Derivative
Find the derivative of the function by first principles (i.e. using the limit definition of the derivative): h(t)=216−th\left(t\right)=2\sqrt{16-t}
Q.\textbf{Q.} Consider again the function f(x)=x2+1f(x)=x^2+1. What is the instantaneous rate of change of ff at x=2x=2? Answer using the definition of the derivative.
Q.\textbf{Q.} Show that f(x)=∣x−2∣−1f(x)=\lvert x-2\rvert-1 is not differentiable at x=2x=2.
Limit Definition of a Derivative: Working with Given Values
Given that f′(2)=4f^{\prime}(2)=4, find lim⁡x→2x−2f(x)−f(2)\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x}-\sqrt{2}}{f(x)-f(2)}.
Limit Definition of a Derivative
Find a function ff and a number aa such that:
f′(a)=lim⁡x→0cos⁡x−1x{f^{\prime}(a)=\lim_{x\rightarrow 0}\frac{\cos{x}-1}{x}}
Limit Definition of a Derivative
Q:\textbf{Q:} Find f′(2)f'(2) if f(x)=(2x+5)12f(x)=(2x+5)^{\frac{1}{2}} using the definition of the derivative.
Limits: Definition with Fractions
Find the derivative of g(x)=12+x\displaystyle g(x)=\frac{1}{2+x} using the definition.
🦊 TRICKY! If f(x)f(x) is differentiable at x=1x=1 and f′(1)=4f'(1)=4 then find the limit lim⁡x→14x3−4x2f(x)−f(1)\displaystyle \lim_{x\rightarrow1}\frac{4x^3-4x^2}{f(x)-f(1)}.
Limit Definition of a Derivative
The given limit represents the derivative of the function f(x)f\left(x\right) at a value aa. Identify the function and the value of aa.
lim⁡h →0 sin⁡(π2+h)−1h\displaystyle\lim_{h\ \to0\ }\frac{\sin\left(\frac{\pi}{2}+h\right)-1}{h}
Limit Definition of a Derivative
Find the derivative of f(x)=13x+1\displaystyle f(x) = \frac{1}{3x + 1} by the definition of the derivative.
Calculate the derivative of the function by first principles (i.e. using the limit definition of the derivative)
h(t)=216−th\left(t\right)=2\sqrt{16-t}
Find f′(2) if f(x)=(2x+5)12f'(2)\ \text{if}\ f(x)=(2x+5)^{\frac{1}{2}} using the definition of the derivative.
Using the definition of the derivative, find f′(x)f'\left(x\right) for f(x)=x2−1f(x)=x^2−1.
Limit definition of the derivative (first principles)
If f(x)=1x,f(x)=\frac{1}{\sqrt{x}},then find the derivative of the function by first principles.
Limit Definition of a Derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=2x+3\displaystyle f(x)=\sqrt{2x+3}
Limit Definition of a Derivative
Evaluate the following limit by recognizing it as a derivative of a function at a certain point.
lim⁡x→1 ln⁡x1−x\lim_{x\to1}\ \frac{\ln x}{1-x}.
Practice: Definition of Derivative
Practice: Definition of Derivative
For which pair of f(x)f\left(x\right) and aa does the equality f(a)=lim⁡x→8 x+1−3x−8\displaystyle f\left(a\right)=\lim_{x\to8}\ \frac{\sqrt{x+1}-3}{x-8} hold?
Limit Definition of Derivatives
Which of the following limits represents a derivative?
i) lim⁡a→0 f(x+a)−f(x)x−a\displaystyle \lim_{a\to0}\ \frac{f\left(x+a\right)-f\left(x\right)}{x-a}
ii) lim⁡x→3 x−3x−3\displaystyle \lim_{x\to3}\ \frac{\sqrt{x-3}}{x-3}
Limit Definition of a Derivative
Using the formal definition of a derivative, the derivative of f(x)=x2f\left(x\right)=\frac{\sqrt{x}}{2} simplifies to
Let f(x)=14−5xf(x) = \frac{1}{4 - 5x} . Use the definition of the derivative to compute f′(3)f'(3) . No credit will be given for the use of any differentiation rule.
Let f(x)=14−5xf(x) = \frac{1}{4 - 5x} . Use the definition of the derivative to compute f′(3)f'(3) . No credit will be given for the use of any differentiation rule.
Limit definition of a derivative
Using the definition of the derivative, solve for the derivative of f(x)f(x) at the point x=2x = 2 :
f(x)=13x+1−1f(x)= \frac{1}{\sqrt{3x + 1} - 1}
Limit Definition of a Derivative
State the definition of the derivative of a function f(x)f(x) at a point x=ax = a .
Definition of the Derivative
Use the definition of the derivative to compute the derivative of
f(x)=x+2x−1f(x) = x + \sqrt{2x - 1}
at x=1x = 1 .
Definition of the Derivative
Carefully state what the definition of the derivative of a function is at the point x=ax = a .
Definition of a Derivative
Using the definition of the derivative, find the derivative of f(x)=(x+2)2+xf(x)=(x+2)^2+x
Limit definition of a derivative
Find f′(x)f^{\prime}(x) using the limit definition of the derivative.
f(x)=1x+3 for x≠−3f(x)=\frac{1}{x+3}\,\text{for}\,x\neq -3
Find f′(x)f^{\prime}(x) using the limit definition of the derivative if f(x)=1x2+3\displaystyle f(x)=\frac{1}{x^2+3}
More Infinite Limits Questions:
Limits: Practice
Find the following limit:
lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Infinite Limits
Determine lim⁡x→π−ln⁡(sin⁡x)\displaystyle\lim_{x \rightarrow\pi^-}\ln(\sin x) .
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Infinite Limits
Q:\bf{Q:} Determine if the function f(x)=eπ+1ln⁡(3x−2)f\left(x\right)=\displaystyle\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
Infinite Limits
Determine lim⁡x→π−ln⁡(sin⁡x)\displaystyle\lim_{x \rightarrow\pi^-}\ln(\sin x) .
Find all asymptotes for f(x)=3x2+x+1x2−1\displaystyle f(x)=\frac{3x^2+x+1}{x^2-1} . Enter as comma-separated values (no spaces in between), smaller value first
Find the exact list of vertical asymptotes of f(x)=x2+4xx2−16\displaystyle f(x)=\frac{x^2+4x}{x^2-16}.
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
General Indeterminate Forms
Evaluate:
lim⁡x→+∞2x+19x2−2x−6\lim_{x \rightarrow +\infty} \frac{2x + 1}{\sqrt{9x^2-2x - 6}}
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Limits at Infinity
lim⁡x→−∞x2+10x−8x+56x−4\displaystyle\lim_{x\rightarrow-\infty}\frac{\sqrt{x^2+10x}-8x+5}{6x-4}
Infinite Limits
lim⁡x→2(x−9)(x+2)x−2\displaystyle \lim_{x\rightarrow2}\frac{(x-9)(x+2)}{x-2}
Limits at Infinity
Find the asymptote(s), if any, of
y=x3−1x2y=\dfrac{x^3 -1}{x^2}
Determine the value of lim⁡t→2+log⁡3(1t−2)\displaystyle \lim_{t \to 2^+}\log_3 \left( \frac{1}{t-2} \right).
Infinite Limits
Q:\bf{Q:} Determine if the function f(x)=eπ+1ln⁡(3x−2)f\left(x\right)=\displaystyle\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Infinite Limits
Determine lim⁡x→π−ln⁡(sin⁡x)\displaystyle\lim_{x \rightarrow\pi^-}\ln(\sin x) .
Find all asymptotes for f(x)=3x2+x+1x2−1\displaystyle f(x)=\frac{3x^2+x+1}{x^2-1} . Enter as comma-separated values (no spaces in between), smaller value first
Find the exact list of vertical asymptotes of f(x)=x2+4xx2−16\displaystyle f(x)=\frac{x^2+4x}{x^2-16}.
Determine if the function h(t)=eπ+1ln⁡(3x−2)h\left(t\right)=\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
Infinite Limits
Find the exact list of vertical asymptotes of
f(x)=x2+4xx2−16f(x)=\frac{x^2+4x}{x^2-16}
Limits: Practice
Find the following limit:
lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}
🌶 TOUGH! Evaluate the limit, lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}.
Determine if the function h(x)=eπ+1ln⁡(3x−2)h\left(x\right)=\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
🦊 TRICKY! Evaluate the limit, lim⁡x→∞x2−10x−x2\displaystyle\lim_{x\rightarrow \infty} \sqrt{x^2-10x}-x^2.
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
More Continuity Questions:
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)\left(a,b\right).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)(a,b).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Use the graph below to find
Continuity
Use the graph below to find
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Practice: Evaluating limits
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.
f(x)f\left(x\right) is discontinuous at x=0
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Find values of aa and bb such that the following function ff is continuous at x=0x=0 and x=1x=1:
f(x)={a(1−cos⁡2xsin⁡(2x)), if x<02x2−x+b, if 0≤x≤1x2+2x−3x2−1, if x>1f(x)=\begin{cases} a\left(\dfrac{\sqrt{1-\cos^2x}}{\sin(2x)}\right)&,\text{ if }x<0\\\\ 2x^2-x+b&,\text{ if }0\le x\le1\\\\ \dfrac{x^2+2x-3}{x^2-1}&,\text{ if }x>1 \end{cases}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={x2+10;x<1−x+7a;x≥1f(x)=\begin{cases} x^2+10;&x<1\\ -x+7a;&x\geq1 \end{cases}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={−2x2+7;x<−43x+a;x≥−4f(x)=\begin{cases} -2x^2+7;&x<-4\\ 3x+a;&x\geq-4 \end{cases}
Continuity and Piecewise Functions
What values of aa and bb make the following function continuous?
f(x)={ax2−4;x<−2ax+b;−2≤x≤3ax2+bx+10;x>3f(x)= \begin{cases} ax^2-4;&&x<-2\\\\ ax+b;&&-2\leq{}x\leq3\\\\ ax^2+bx+10;&&x>3 \end{cases}
Continuity
Determine whether the following function is continuous at x=0x = 0 using the definition of continuity. You must full justify your solution.
g(x)={x2+4−2x , x<00 , x≥0\displaystyle g(x) = \begin{cases} \frac{\sqrt{x^2 + 4} - 2}{x} & \text{ , } x < 0 \\ 0 & \text{ , } x \geq 0 \end{cases}
Continuity
Let f(x)f(x) be a continuous function on the open interval (a,b)(a, b). Which of the following four statements are always true.
Select all that apply.
Continuity
Where is f(x)=cos⁡(πx2)1−x2f(x) = \frac{\cos\left(\frac{\pi x}{2} \right)}{\sqrt{1 - x^2}} continuous?
f(x)={a2x2+1if x<1x+4if x=1−5(a+x)if x>1f\left(x\right)= \begin{cases} a^2x^2+1&if\space x<1\\ x+4 & if \space x=1\\ -5(a+x)&if~x>1 \end{cases}
a) For what value(s) of aa does lim⁡x→1f(x)\displaystyle \lim_{x\to1}f(x) exist?
b) For what value(s) of aa is f(x)f(x) continuous at x=1x=1?
Continuity
Use the graph below to find
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Practice: Continuity of Piecewise Functions
Consider the function
f(x)={ln⁡(x2)+xif  x<−1cos⁡πx2if  −1≤x<12exif  x≥12f\left(x\right)= \begin{cases} \ln(x^2)+x & \text{if }\space x<-1\\ \cos \frac{{\pi}x}{2}&\text{if }\space -1\le x<\frac{1}{2}\\ e^{x}&\text{if }\space x\ge \frac{1}{2} \end{cases}
Limits: Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1 if x≠−2, x≠24 if x=25 if x=−2f(x) = \begin{cases} \frac{x^3-8}{x^2-4}+1 & \text{ if } x\neq -2, \, x \neq 2 \\ 4 & \text{ if } x = 2 \\ 5 & \text{ if } x = -2 \end{cases}
Limits: Continuity
Find A and B so that the following function is continuous at 0 and 1:
f(x)={A(1−cos⁡2xsin⁡2x)x<02x2−x+B0≤x≤1x2+2x−3x2−1x>1f(x) = \begin{cases} A\left(\frac{\sqrt{1-\cos^2 x}}{\sin 2x}\right) & x < 0 \\ 2x^2-x+B & 0 \leq x \leq 1 \\ \frac{x^2+2x-3}{x^2-1} & x > 1 \end{cases}
Enter your answer as A,B. For instance, if you think A=0, B=1, enter 0,1
If f(x)f(x) is continuous at x=−2x=-2 and if lim⁡x→−2(2f(x)−x)2∣f(x)∣2+1=4\lim\limits_{x\rightarrow -2}\frac{(2f(x)-x)^2}{|f(x)|^2+1}=4, then
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Find all number(s) C that make(s) f(x) continuous.
f(x)={x−C1+Cx≤02x2+Cx>0f(x)=\begin{cases}\displaystyle\frac{x-C}{1+C}&x\leq0\\2x^2+C&x>0\end{cases}
Discuss where the function is continuous: f(x)=sin⁡(x+cos⁡x)f(x)=\sin(x+\cos x).
If f(x)f(x) is continuous at x=2x=2 and if lim⁡x→23f(x)+2xf(x)−2=4\lim\limits_{x\rightarrow 2}\frac{3f(x)+2}{xf(x)-2}=4, then
Continuity
Find aa such that f(x)f(x) is continuous at x=1x = 1
f(x)={4x2−3ln⁡xx≥13x3−4axx<1f(x)= \begin{cases} 4x^2 - 3\ln{x} & x \ge 1 \\ 3x^3 - 4ax & x < 1 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuity
Discuss where the following function is continuous:
h(x)=log⁡(x+1−2)h(x)=\log(\sqrt{x+1}-2)
Consider f(x)=sin⁡2xx\displaystyle f(x)=\frac{\sin{2x}}{x} . Does lim⁡x→0sin⁡2xx\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}}{x} exist? Is ff continuous at x=0x=0?
When is the following function continuous: g(x)=x2−1x2−1\displaystyle g(x)=\frac{x^2-1}{x^2-1}?
Where is the following function continuous?
f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
Find aa,bb such that f(x)f(x) is continuous everywhere.
f(x)={x2+x+ax>1x−b0≤x≤1x3−4x+5x<0f(x)=\begin{cases}x^2+x+a&x>1\\x-b&0\leq x\leq1\\x^3-4x+5&x<0\end{cases}
🦊 TRICKY! Find A > 0 and B so that the following function is continuous at x = 0 and x = 1:
f(x)={x2tan⁡2Axifx<03x2+2x+Bif0≤x≤13x2−7x−20x2−3x−4ifx>1f(x)=\begin{cases} \frac{x^2}{\tan^2Ax}&\text{if}&x<0\\ 3x^2+2x+B&\text{if}&0\leq x\leq 1\\ \frac{3x^2-7x-20}{x^2-3x-4}&\text{if}&x>1\end{cases}
Continuity
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Limits: Continuity of piecewise functions
Identify the type of discontinuity (or state the function is continuous) at x=5,10,15x=5, 10, 15. Is there another point of discontinuity in the domain other than at the stated x-values?
f(x)={x+3x−6x−6+7ln⁡(15−x)sin⁡xif−∞<x≤5if 5<x≤10if10<x<15if15≤x<∞f(x)=\begin{cases} x+3\\ \frac{x-6}{x-6}+7\\ \ln(15-x)\\ \sin x \end{cases}\begin{array}{l} if -\infty < x \leq 5\\ if\ 5 < x \leq 10\\ if 10 < x< 15\\ if 15 \leq x < \infty \end{array}
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Continuity
Find the value of aa such that the function
f(x)={x2+1if x≤−3ax−1if x>−3f\left(x\right)= \begin{cases} x^2+1&if\space x\le -3\\ ax-1 & if \space x>-3 \end{cases}
is continuous everywhere.
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right), which is defined only defined on [−5,9][-5,9]. Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\dfrac{1}{\ln\left(f\left(x\right)\right)} is continuous.
Practice: Evaluating limits
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.
f(x)f\left(x\right) is discontinuous at x=0
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Each question is worth 3 marks.
Continuity
What value of aa makes the following function continuous?
f(x)={2x+4,x≤2ax2+x−2,2<xf(x) = \left \{ \begin{array}{ll} 2^x + 4, & x \leq 2\\ ax^2 + x - 2, & 2 < x \end{array} \right .
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
When is the following function continuous?
e3x+1sin⁡(πx)\displaystyle \frac{e^{3x}+1}{\sin(\pi x)}