Limits of Trigonometric Functions

Evaluate $\underset{x\to 0}{\lim } \frac{\sqrt{7}-\sqrt{6+\cos x}}{x\sin x}$

Evaluate $\underset{x\to 0}{\lim }\frac{\cos 2x-1}{\cos x-1}$.

Evaluate: $\underset{x\to 0}{\lim }\frac{\sin ax+bx}{ax+\sin bx}$ where $a,b,a+b\ne 0$

Find the value of $\underset{x\to 0}{\lim }\frac{\sin 3x}{x}$

$\underset{x\to 0}{\lim }\frac{\sin x}{\sqrt{x}}$ is _____

If $\alpha >\beta >0$ are the roots of the equation $a{x}^2+bx+1=0$, and
$\underset{x\to \frac{1}{\alpha }}{\lim }{\left(\frac{1-\cos \left(x^2+bx+a\right)}{2(1-\alpha x{)}^2}\right)}^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta }-\frac{1}{\alpha }\right)$, then $k$ is equal to

$\underset{n\to \infty }{\lim }\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right)....\left(2^{\frac{1}{2}}-2^{\frac{1}{2n+1}}\right)\right\}$ is equal to

Evaluate: $\underset{x\to 0}{\lim }\frac{{\sin }^{-1}x}{\sin x}$

If $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$, then $\underset{x\to \frac{1}{\alpha }}{\lim }\left(\frac{1-\cos \left(c{x}^2+bx+a\right)}{2{\left(1-\alpha x\right)}^2}\right)$ is

Evaluate $\underset{x\to 0}{\lim }\frac{\cos 2x-1}{\cos x-1}$.

Evaluate: $\underset{x\to 0}{\lim }\frac{\sin ax+bx}{ax+\sin bx}$ where $a,b,a+b\ne 0$

Evaluate: $\underset{x\to 0}{\lim }\frac{x}{{\sin }^{-1}x}$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ then $\underset{x\to \frac{1}{\alpha }}{\lim }\sqrt{\frac{1-\cos \left(c{x}^2+bx+a\right)}{2(1-\alpha x{)}^2}}$

Prove that $\underset{x\to 0}{\lim }\frac{\sin x°}{x}=\frac{{\displaystyle \mathrm{\pi }}}{180}$

Value of limit,

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{4\sqrt{2}\left(\sin 3x+\sin x\right)}{\left(2\sin 2x\sin {\displaystyle \frac{3x}{2}}+\cos {\displaystyle \frac{5x}{2}}\right)-\left(\sqrt{2}+\sqrt{2}\cos 2x+\cos {\displaystyle \frac{3x}{2}}\right)}$

Evaluate: $\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\sqrt{\cos x}-\sqrt{\sin x}}{x-{\displaystyle \frac{\mathrm{\pi }}{4}}}$

$\underset{x\to 0}{\lim }\frac{\sin \alpha x-\sin \beta x}{e^{\alpha x}-e^{\beta x}}$

The value of $\underset{x\to 0}{\lim }\frac{2\cos x-\cos 3x-\cos 5x}{x^2}$ is equal to

Find $\underset{x\to 0}{\lim }\frac{{\sin }^{2}4x}{x^2}$

$\begin{array}{rl}& \underset{\mathrm{x}\to \mathrm{\pi }/4}{lim}\frac{{\cot }^3\mathrm{x}-\tan \mathrm{x}}{\cos (\mathrm{x}+\mathrm{\pi }/4)}=?\end{array}$