Limits of Trigonometric functions Questions and solutions

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

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

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

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

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

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

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

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

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

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

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

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

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

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