# Limits of Trigonometric functions Questions and solutions

The limits problems involving the trigonometric functions appear in calculus. So, the limits of trigonometric functions worksheet is given here for you and it consists of simple to tough trigonometric limits examples with answers for your practice, and also solutions to learn how to find the limits of trigonometric functions in possible different methods by the trigonometric limits formulas.

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \Big(\dfrac{\sin{x}}{x}\Big)^{\dfrac{1}{x^2}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{mx}}{1-\cos{nx}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\log_{\displaystyle e}{\big(\cos{(\sin{x})}\big)}}{x^2}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,\Large \frac{\pi}{4}}{\normalsize \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sin^3{x}}{\sin{x}-\tan{x}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\log_{e}{(\cos{x})}}{\sqrt[\Large 4]{1+x^2}-1}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{e^x-e^{x\cos{x}}}{x+\sin{x}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{(2x)}}{x^2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Big(1+\sin{x}\Big)^{\Large \frac{1}{x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{(e^{-3x+2}-e^2)\sin{\pi x}}{4x^2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \dfrac{\cos{\Big(\dfrac{\pi}{x}\Big)}}{x-2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x\tan{2x}-2x\tan{x}}{(1-\cos{2x})^2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, \frac{\pi}{2}}{\normalsize \dfrac{\cos{x}}{\frac{\pi}{2}-x}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, \pi}{\normalsize \dfrac{\sqrt{2+\cos{x}}-1}{(\pi-x)^2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\tan{x}-\sin{x}}{x^3}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{(1-\cos{2x})(3+\cos{x})}{x\tan{4x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, \frac{\pi}{2}}{\normalsize \dfrac{1+\cos{2x}}{(\pi-2x)^2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \sqrt[x^3]{1-x+\sin{x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{\sin{4x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, \pi}{\normalsize \dfrac{x-\pi}{\sin{x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x-\sin{x}}{x^3}}$

Find $\large \displaystyle \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{(\pi\cos^2{x})}}{x^2}}$

Evaluate $\displaystyle \lim_{x \,\to\, 0}{\dfrac{\sin{2x}+3x}{4x+\sin{6x}}}$

$\displaystyle \large \lim_{x \,\to\, \tan^{-1}{3}} \normalsize {\dfrac{\tan^2{x}-2\tan{x}-3}{\tan^2{x}-4\tan{x}+3}}$

$\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{1-\cos{6x}}{1-\cos{7x}}$

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x^3\sin{x}}{{(\sec{x}-\cos{x})}^2}}$

$\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{3\sin{\pi x}-\sin{3\pi x}}{{(x-1)}^3}}$

$\displaystyle \large \lim_{x \,\to\, \pi} \, \normalsize \dfrac{1-\cos{7(x-\pi)}}{5{(x-\pi)}^2}$

$\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{1-\sqrt{1-\tan x}}{\sin x}$

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{2\sin{x}-\sin{2x}}{x^3}}$

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{\cos{\sqrt{x}}-\cos{\sqrt{a}}}{x-a}}$

$\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{\sin{(x-1)}}{x^2-1}}$

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{1-\cos{x}\sqrt{\cos{2x}}}{x^2}}$

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\cos{3x}-\cos{4x}}{x\sin{2x}}}$

$\displaystyle \large \lim_{\theta \,\to\, 0}{\normalsize \dfrac{\sin{5\theta}-\sin{3\theta}}{\theta}}$

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Jun 26, 2023

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