Evaluate $\lim _{x \rightarrow 3} \frac{x^2-9}{x-3}$.

Evaluate $\lim _\limits{x \rightarrow 1 / 2} \frac{4 x^2-1}{2 x-1}$.

Evaluate $\lim _\limits{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$.

Evaluate $\lim _\limits{x \rightarrow 0} \frac{(x+2)^{1 / 3}-2^{1 / 3}}{x}$.

Evaluate $\lim _\limits{x \rightarrow 0} \frac{(1+x)^6-1}{(1+x)^2-1}$.

Evaluate $\lim _\limits{x \rightarrow a} \frac{(2+x)^{5 / 2}-(a+2)^{5 / 2}}{x-a}$

Evaluate $\lim _\limits{x \rightarrow 1} \frac{x^4-\sqrt{x}}{\sqrt{x}-1}$.

Evaluate $\lim _\limits{x \rightarrow 2} \frac{x^2-4}{\sqrt{3 x-2}-\sqrt{x+2}}$

Evaluate $\lim _\limits{x \rightarrow \sqrt{2}} \frac{x^4-4}{x^2+3 \sqrt{2} x-8}$

Evaluate $\lim _\limits{x \rightarrow 1} \frac{x^7-2 x^5+1}{x^3-3 x^2+2}$

$$\text { Evaluate } \lim _\limits{x \rightarrow 0} \frac{\sqrt{1+x^3}-\sqrt{1-x^3}}{x^2} .$$

Evaluate $\lim _\limits{x \rightarrow-3} \frac{x^3+27}{x^5+243}$

Evaluate $\lim _\limits{x \rightarrow 1 / 2}\left(\frac{8 x-3}{2 x-1}-\frac{4 x^2+1}{4 x^2-1}\right)$

Find the value of $n$, if $\lim _\limits{x \rightarrow 2} \frac{x^n-2^n}{x-2}=80, n \in N$.

Evaluate $\lim _\limits{x \rightarrow 0} \frac{\sin 3 x}{\sin 7 x}$

Eavaluate $\lim _\limits{x \rightarrow 0} \frac{\sin ^2 2 x}{\sin ^2 4 x}$.

Evaluate $\lim _\limits{x \rightarrow 0} \frac{1-\cos 2 x}{x^2}$.

Evaluate $\lim _\limits{x \rightarrow 0} \frac{2 \sin x-\sin 2 x}{x^3}$

Evaluate $\lim _\limits{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$

Evaluate $\lim _\limits{x \rightarrow \pi / 3} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$.

Evaluate $\lim _\limits{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{x-\frac{\pi}{4}}$.

Evaluate $\lim _\limits{x \rightarrow \pi / 6} \frac{\sqrt{3} \sin x-\cos x}{x-\frac{\pi}{6}}$.

Evaluate $\lim _\limits{x \rightarrow 0} \frac{\sin 2 x+3 x}{2 x+\tan 3 x}$

Evaluate $\lim _\limits{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$

Evaluate $\lim _\limits{x \rightarrow \pi / 6} \frac{\cot ^2 x-3}{\operatorname{cosec} x-2}$

Evaluate $\lim _\limits{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^2 x}$

Evaluate $\lim _\limits{x \rightarrow 0} \frac{\sin x-2 \sin 3 x+\sin 5 x}{x}$

If $\lim _\limits{x \rightarrow 1} \frac{x^4-1}{x-1}=\lim _\limits{x \rightarrow k} \frac{x^3-k^3}{x^2-k^2}$, then find the value of $k$.

$\frac{x^4+x^3+x^2+1}{x}$

$$ \left(x+\frac{1}{x}\right)^3 $$

$$ (3 x+5)(1+\tan x) $$

$(\sec x-1)(\sec x+1)$

$\frac{3 x+4}{5 x^2-7 x+9}$

$\frac{x^5-\cos x}{\sin x}$

$\frac{x^2 \cos \frac{\pi}{4}}{\sin x}$

$(a x^2+\cot x)(p+q \cos x)$

$\frac{a+b \sin x}{c+d \cos x}$

$(\sin x+\cos x)^2$

$(2 x-7)^2(3 x+5)^3$

$x^2 \sin x+\cos 2 x$

$\sin^3x\cos^3x$

$\frac{1}{a x^2+b x+c}$

$\cos \left(x^2+1\right)$

$$ \frac{a x+b}{c x+d} $$

$x^{2/3}$

$x\cos x$

$\lim _\limits{y \rightarrow 0} \frac{(x+y) \sec (x+y)-x \sec x}{y}$

$\lim _\limits{x \rightarrow 0} \frac{\sin (\alpha+\beta) x+\sin (\alpha-\beta) x+\sin 2 \alpha x}{\cos 2 \beta x-\cos 2 \alpha x} \cdot x$

$$\lim _\limits{x \rightarrow \pi / 4} \frac{\tan ^3 x-\tan x}{\cos \left(x+\frac{\pi}{4}\right)}$$

$\lim _\limits{x \rightarrow \pi} \frac{1-\sin \frac{x}{2}}{\cos \frac{x}{2}\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)}$

Show that $\lim _\limits{x \rightarrow \pi / 4} \frac{|x-4|}{x-4}$ does not exist,

If $f(x)=\left\{\begin{array}{cl}\frac{k \cos x}{\pi-2 x}, & \text { when } x \neq \frac{\pi}{2} \\ 3, & \text { when } x=\frac{\pi}{2}\end{array}\right.$ and $\lim _\limits{x \rightarrow \pi / 2} f(x)=f\left(\frac{\pi}{2}\right)$, then find the value of $k$.

If $f(x)=\left\{\begin{array}{ll}x+2, & x \leq-1 \\ c x^2, & x>-1\end{array}\right.$, then find $c$ when $\lim _\limits{x \rightarrow-1} f(x)$ exists.

$\lim _\limits{x \rightarrow \pi} \frac{\sin x}{x-\pi}$ is equal to

$\lim _\limits{x \rightarrow 0} \frac{x^2 \cos x}{1-\cos x}$ is equal to

$\lim _\limits{x \rightarrow 0} \frac{(1+x)^n-1}{x}$ is equal to

$\lim _\limits{x \rightarrow 1} \frac{x^m-1}{x^n-1}$ is equal to

$\lim _\limits{\theta \rightarrow 0} \frac{1-\cos 4 \theta}{1-\cos 6 \theta}$ is equal to

$\lim _\limits{x \rightarrow 0} \frac{\operatorname{cosec} x-\cot x}{x}$ is equal to

$\lim _\limits{x \rightarrow 0} \frac{\sin x}{\sqrt{x+1}-\sqrt{1-x}}$ is equal to

$\lim _\limits{x \rightarrow \pi / 4} \frac{\sec ^2 x-2}{\tan x-1}$ is

$\lim _\limits{x \rightarrow 1} \frac{(\sqrt{x}-1)(2 x-3)}{2 x^2+x-3}$ is equal to

If $f(x)=\left\{\begin{array}{cc}\frac{\sin [x]}{[x]}, & {[x] \neq 0} \\ 0, & {[x]=0}\end{array}\right.$, where [.] denotes the greatest integer function, then $\lim _\limits{x \rightarrow 0} f(x)$ is equal to

$\lim _\limits{x \rightarrow 0} \frac{|\sin x|}{x}$ is equal to

If $f(x)=\left\{\begin{array}{ll}x^2-1, & 0< x< 2 \\ 2 x+3, & 2 \leq x < 3\end{array}\right.$, then the quadratic equation whose roots are $\lim _\limits{x \rightarrow 2^{-}} f(x)$ and $\lim_\limits{x \rightarrow 2^{+}}{f(x)}$ is

$\lim _{x \rightarrow 0} \frac{\tan 2 x-x}{3 x-\sin x}$ is equal to

If $f(x)=x-[x], \in R$, then $f^{\prime}\left(\frac{1}{2}\right)$ is equal to

If $y=\sqrt{x}+\frac{1}{\sqrt{x}}$, then $\frac{d y}{d x}$ at $x=1$ is equal to

If $f(x)=\frac{x-4}{2 \sqrt{x}}$, then $f^{\prime}(1)$ is equal to

If $y=\frac{1+\frac{1}{x^2}}{1-\frac{1}{x^2}}$, then $\frac{d y}{d x}$ is equal to

If $y=\frac{\sin x+\cos x}{\sin x-\cos x}$, then $\frac{d y}{d x}$ at $x=0$ is equal to

If $y=\frac{\sin (x+9)}{\cos x}$, then $\frac{d y}{d x}$ at $x=0$ is equal to

If $f(x)=1+x+\frac{x^2}{2}+\ldots+\frac{x^{100}}{100}$, then $f^{\prime}(1)$ is equal to

If $f(x)=\frac{x^n-a^n}{x-a}$ for some constant $a$, then $f^{\prime}(a)$ is equal to

If $f(x)=x^{100}+x^{99}+\ldots+x+1$, then $f^{\prime}(1)$ is equal to

If $f(x)=1-x+x^2-x^3+\ldots-x^{99}+x^{100}$, then $f^{\prime}(1)$ is equal to

If $f(x)=\frac{\tan x}{x-\pi}$, then $\lim _\limits{x \rightarrow \pi} f(x)=$

$\lim _\limits{x \rightarrow 0}\left(\sin m x \cot \frac{x}{\sqrt{3}}\right)=2$, then $m=$ ...........

If $y=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots$, then $\frac{d y}{d x}=$ .........

$\lim _\limits{x \rightarrow 3^{+}} \frac{x}{[x]}=$ .........