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Trigonometric Limits

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Wize University Calculus 1 Textbook > Limits

The Fundamental Trig Limit

2 Activities

Wize University Calculus 1 Textbook > Limits

Infinite Limits

3 Activities

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Practice: Trig Limits

Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
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}.
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.
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}}
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}
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}}
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)?
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}.