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Trigonometric Limits Problems and Solutions

List of trigonometric limits problems with step by steps solutions for those want to learn and practice trigonometric functions basis limits problems.

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

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

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

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

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

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

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

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

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

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

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

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

$$\lim_{\theta \to 0} \dfrac{\sin 5\theta -\sin 3\theta}{\theta}$$