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Practice: Extracting Highest Power Indeterminate Forms

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Wize University Calculus 1 Textbook > Limits

Limits at Infinity

6 Activities

  • 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.

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}.
Infinite Limits and Continuity
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)?
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: 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}