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Limits and Continuity

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Wize University Calculus 1 Textbook > Limits

The Fundamental Trig Limit

2 Activities

Wize University Calculus 1 Textbook > Limits

Computing Limits by Multiplying by the Conjugate

3 Activities

Wize University Calculus 1 Textbook > Limits

Limits at Infinity

6 Activities

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}}


c) lim⁡x→0sin⁡(3x)sin⁡(5x)sin⁡(x)sin⁡(8x)\displaystyle \lim_{x \rightarrow 0} \frac{\sin(3x) \sin(5x)}{ \sin(x) \sin(8x)}
More The Fundamental Trig Limit Questions:
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Fundamental trigonometric limits
Evaluate lim⁡x→0tan⁡x−sin⁡xxcos⁡x\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}.
Fundamental trigonometric limits
Evaluate lim⁡x→0tan⁡x−sin⁡xxcos⁡x\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}.
Fundamental trigonometric limits
Determine lim⁡x→0x2tan⁡2 2x\lim\limits_{x\rightarrow0}\frac{x^2}{\tan^2\ 2x}.
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Practice: Using Known Trig Limits Indeterminate Forms
Behaviour of Trigonometric Functions: Evaluate lim⁡x→0sin⁡7x4x\lim_{x\to0}\limits\dfrac{\sin 7x}{4x}
Limits: Trigonometric Functions
Evaluate lim⁡x→3x2−2x−3πsin⁡(x−3)\displaystyle\lim_{x\to 3}\dfrac{x^2-2x-3}{\pi\sin(x-3)}
Limits: Trigonometric Functions
Evaluate lim⁡x→0tan⁡2x4x(x2−1)\displaystyle\lim_{x\to 0}\dfrac{\tan 2x}{4x(x^2-1)}
Evaluate lim⁡x→0sin⁡7x4x\displaystyle \lim_{x\rightarrow 0}\frac{\sin{7x}}{4x}.
Fundamental trigonometric limits
Evaluate lim⁡x→0tan⁡x−sin⁡xxcos⁡x\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}.
Trigonometric Limits: Combining techniques
Evaluatelim⁡x→0xln⁡(1x+2x)e1+tan⁡ x+ln⁡ x(1+arctan⁡ x)sin⁡ x+xcos⁡ x\lim\limits_{x\rightarrow0}\frac{x\ln(\frac{1}{x+2^x})e^{{1+\tan\ x+\ln\ x}}}{(1+\arctan\ x)\sin\ x+x\cos\ x}
🌶️ TOUGH! Evaluate lim⁡x→πsin⁡xx−π\lim\limits_{x\rightarrow \pi}\frac{\sin x}{x-\pi}. Hint: use the trigonometric identity sin⁡(θ+π)=−sin⁡(θ)\sin(\theta+\pi)=-\sin(\theta).
Determine lim⁡x→0x2tan⁡23x\lim\limits_{x\rightarrow 0}\frac{x^2}{\tan^23x}.
Evaluate the following limit.
lim⁡x→0tan⁡(ax)sin⁡x\displaystyle \lim_{x\rightarrow 0}\frac{\tan(ax)}{\sin x}
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?
Practice: Fundamental trigonometric limits
Evaluatelim⁡x→1−sin⁡1−x1−x2\lim\limits_{x\rightarrow1^-}\frac{\sin\sqrt{1-x}}{\sqrt{1-x^2}}
Practice: Fundamental trigonometric limits
Part a)
Evaluatelim⁡x→0sin⁡(2πx)sin⁡(3πx)\lim\limits_{x\rightarrow0}\frac{\sin(2\pi x)}{\sin(3\pi x)}
Part b)
Fundamental trigonometric limits
Determine lim⁡x→0x2tan⁡2 2x\lim\limits_{x\rightarrow0}\frac{x^2}{\tan^2\ 2x}.
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Fundamental Trig Limit
Evaluate the following limit:
lim⁡x→0sin⁡(3x)cos⁡(2x)sin⁡(5x)\lim_{x\rightarrow0}\frac{\sin(3x)\cos(2x)}{\sin(5x)}
Each question is worth 3 marks.
More Computing Limits by Multiplying by the Conjugate Questions:
Limits: Multiplying by the Conjugate
lim⁡x→∞x2+2x−x2+1\lim_{x\rightarrow\infty}\sqrt{x^2+2x}-\sqrt{x^2+1}
Limit Techniques: Multiply by the Conjugate
lim⁡x→π21−sin⁡xcos⁡x\displaystyle \lim_{x\rightarrow\frac{\pi}{2}}\frac{1-\sin{x}}{\cos{x}}
∞ - ∞ Limits with Square Root
lim⁡x→∞ x2+2x−x2−3x=\displaystyle\lim_{x\to\infty}\ \sqrt{x^2+2x}-\sqrt{x^2-3x}=
∞ - ∞ Limits with Square Root
lim⁡x→∞ x2+2x−x2−3x=\displaystyle\lim_{x\to\infty}\ \sqrt{x^2+2x}-\sqrt{x^2-3x}=
Evaluating limits: Multiplying by the Conjugate
Practice: Evaluating limits
Determine the value of lim⁡x →0 x+1−1−xx \lim_{x\ \rightarrow0}\ \frac{\sqrt{x+1}-\sqrt{1-x}}{x}\
Find the following limit: lim⁡x→3x2−9x−3\displaystyle\lim_{x\rightarrow3}\frac{x^2-9}{\sqrt{x}-\sqrt{3}}
Practice: Limit at Infinity with Roots
Q.\textbf{Q.} Evaluate the limit: lim⁡x→∞x2−2−x2+x\displaystyle \lim_{x\rightarrow \infty}\sqrt{x^2-2}-\sqrt{x^2+x}
Limit at Infinity with Roots
Evaluate lim⁡x→∞9x2+2x−3x\displaystyle \lim_{x\rightarrow\infty}\sqrt{9x^2+2x}-3x.
Limit with Unknown Coefficients
Find constants aa and bb such that lim⁡x→0ax+b−2x=1\displaystyle\lim_{x\rightarrow 0}\frac{\sqrt{ax+b}-2}{x}=1.
Square Roots: Limits by Multiplying by the Conjugate
Q.\textbf{Q.} Evaluate lim⁡x→2x+7−3x−2\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x+7}-3}{x-2}
Limits by multiplying by the conjugate
lim⁡t→13+t−2t3−1\lim_{t\rightarrow1}\dfrac{\sqrt{3+t}-2}{t^3-1}
Evaluate lim⁡x→9x+7−42x−18\displaystyle\lim_{x\to9} \frac{\sqrt{x+7}-4}{2x-18}
l'Hopital's Rule
Evaluate
lim⁡x→∞x2+1−x\lim\limits_{x\to\infty} \sqrt{x^2+1}-x
Practice: Rationalizing Indeterminate Forms
Rationalizing: Evaluate lim⁡x→3x2−9x−3\lim_{x\to3}\limits\dfrac{x^2 - 9}{\sqrt x - \sqrt 3}
Limits: Multiplying Conjugates
Find constants a and b such that
lim⁡x→0ax+b−2x=1\lim\limits_{x\to 0} \frac{\sqrt{ax+b}-2}{x}=1
Limits: Multiplying Conjugates
Evaluate the limit:
lim⁡x→2x+7−3x−2\lim\limits_{x \to 2} \frac{\sqrt{x+7} - 3}{x-2}
The value of the limit lim⁡x→1x−1x2−3x+2\lim_{x\to1}\frac{\sqrt{x}-1}{x^2-3x+2} is
Limits by Multiplying by the Conjugate
The value of the limits lim⁡x→1x−1x2−4x+3\lim\limits_{x\rightarrow1}\frac{\sqrt{x}-1}{x^2-4x+3}
Determine the value of lim⁡x→24x−1−7x−2 \displaystyle \lim_{x \to 2} \frac{\sqrt{4x-1}-\sqrt{7}}{x-2}.
The value of the limit lim⁡x→1x−1x2−3x+2\lim_{x\to1}\frac{\sqrt{x}-1}{x^2-3x+2} is
Limit with Unknown Coefficients
Find constants aa and bb such that lim⁡x→0ax+b−2x=1\displaystyle\lim_{x\rightarrow 0}\frac{\sqrt{ax+b}-2}{x}=1.
Square Roots: Limits by Multiplying by the Conjugate
Q.\textbf{Q.} Evaluate lim⁡x→2x+7−3x−2\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x+7}-3}{x-2}
Practice: Limit at Infinity with Roots
Q.\textbf{Q.} Evaluate the limit: lim⁡x→∞x2−2−x2+x\displaystyle \lim_{x\rightarrow \infty}\sqrt{x^2-2}-\sqrt{x^2+x}
Find the following limit: lim⁡x→3x2−9x−3\displaystyle\lim_{x\rightarrow3}\frac{x^2-9}{\sqrt{x}-\sqrt{3}}
The value of the limit lim⁡x→1x−1x2−3x+2\lim_{x\to1}\frac{\sqrt{x}-1}{x^2-3x+2} is
Limit at Infinity with Roots
Evaluate lim⁡x→∞9x2+2x−3x\displaystyle \lim_{x\rightarrow\infty}\sqrt{9x^2+2x}-3x.
Practice: Multiply by conjugate
Evaluatelim⁡x→−∞(x2+3x+x)\lim\limits_{x\rightarrow-\infin}(\sqrt{x^2+3x}+x)
Evaluate the limit
lim⁡x→23x−1−5x−2\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{3x-1}-\sqrt{5}}{x-2}
Evaluate lim⁡t→13+t−2t3−1\displaystyle \lim_{t\rightarrow 1}\frac{\sqrt{3+t}-2}{t^3-1}.
Determine the value of lim⁡x→∞4x4+4x2+1−2x2\displaystyle \lim_{x\rightarrow \infty} \sqrt{4x^4+4x^2+1}-2x^2.
🌶️ TOUGH! Evaluate the limit, lim⁡x→−∞x2−6x+x\displaystyle\lim_{x\rightarrow-\infty}\sqrt{x^2-6x}+x.
Find the following limit: lim⁡x→+∞x2−8x−x\displaystyle \lim_{x\rightarrow+\infty}\sqrt{x^2-8x}-x.
Evaluate lim⁡x→5x−5x2−4x−5\displaystyle \lim_{x\rightarrow 5}\frac{\sqrt{x}-\sqrt{5}}{x^2-4x-5}.
Limits: Combining techniques
Solve the limit lim⁡x→39x−6x−1x2−5−2\displaystyle \lim\limits_{x\to 3} \frac{\frac{9}{x}-\frac{6}{x-1}}{\sqrt{x^2-5}-2}.
lim⁡x→5x−5x2−4x−5\displaystyle\lim_{x\rightarrow5}\dfrac{\sqrt{x}-\sqrt{5}}{x^2-4x-5}
lim⁡x→∞x2+2x−x2+1\displaystyle\lim_{x\rightarrow\infty}\sqrt{x^2+2x}-\sqrt{x^2+1}
Evaluating limits: Multiplying by the Conjugate
Practice: Evaluating limits
Determine the value of lim⁡x →0 x+1−1−xx \lim_{x\ \rightarrow0}\ \frac{\sqrt{x+1}-\sqrt{1-x}}{x}\
∞ - ∞ 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}=
Practice: $\frac{0}{0}$ Limit with Square Root
Practice: 0/0 Limit with Square Root
lim⁡x→0 xx+4−4−x=\displaystyle\lim_{x\to0}\ \frac{x}{\sqrt{x+4}-\sqrt{4-x}}=
Limit Techniques: Multiply by the Conjugate
lim⁡x→π21−sin⁡xcos⁡x\displaystyle \lim_{x\rightarrow\frac{\pi}{2}}\frac{1-\sin{x}}{\cos{x}}
Limits by multiplying by the conjugate
lim⁡x→−∞(9x2−15x+3+3x)\displaystyle\lim_{x\rightarrow -\infty}\big(\sqrt{9x^2-15x+3}+3x\big)
Limits by multiplying by the conjugate
lim⁡x→5x−5x2−4x−5\displaystyle\lim_{x\rightarrow5}\dfrac{\sqrt{x}-\sqrt{5}}{x^2-4x-5}
Limits: Multiplying by the Conjugate
lim⁡x→∞x2+2x−x2+1\lim_{x\rightarrow\infty}\sqrt{x^2+2x}-\sqrt{x^2+1}
Limits by Multiplying by the Conjugate
lim⁡x→0x2+x−2−x\displaystyle\lim_{x\rightarrow0}\dfrac{x}{\sqrt{2+x}-\sqrt{2-x}}
Practice: L'hopital's rule
Q.\textbf{Q.} Evaluate the following limits.
1. lim⁡x→0+cos⁡xcot⁡x\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
2. lim⁡x→∞[x2+1−x]\displaystyle \lim_{x\rightarrow \infty}\left[\sqrt{x^2+1}-x\right]
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: 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 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}