List - I		List - II
A)	$\underset{x\to 0}{lim}{\left(\frac{\sin x}{x}\right)}^{\frac{1}{x^2}}$	1)	$e^{\frac{-1}{6}}$
B)	$\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{\frac{1}{x^2}}$	2)	$e^{\frac{1}{3}}$
C)	$\underset{x\to 0}{lim}(\cos x{)}^{\frac{1}{\tan x}}$	3)	$e^{\frac{2}{\pi }}$
D)	$\underset{x\to a}{lim}{(2-\frac{x}{a})}^{\mathrm{Tan}\left(\frac{\pi x}{2a}\right)}$	4)	1

Option(A)$\Rightarrow$$\underset{x\to 0}{lim}{\left(\frac{\sin x}{x}\right)}^{\frac{1}{x^2}}$

$$\begin{gathered} L=\underset{x\to 0}{lim}{\left(\frac{\sin x}{x}\right)}^{\frac{1}{x^2}} \\ Apply \log on both sides \\ \log L=\underset{x\to 0}{lim}\frac{\log \left(\frac{\sin x}{x}\right)}{x^2} \\ Apply L\text{'}hospital rule \\ \log L=\underset{x\to 0}{lim} \left[\frac{\frac{x}{\sin x}\frac{x\cos x-\sin x}{x^2}}{2x}\right] \\ \log L=\underset{x\to 0}{lim} \left[\frac{x}{\sin x}\cdot \frac{x\cos x-\sin x}{2x^3}\right] \\ \log L=\underset{x\to 0}{lim} \left[\frac{x\cos x-\sin x}{2x^2\sin x}\right] \\ Apply L\text{'}hospital rule \\ \log L=\frac{1}{2}\underset{x\to 0}{lim} \left[\frac{-x\sin x}{2x\sin x+x^2\cos x}\right] \\ \log L=\frac{1}{2}\underset{x\to 0}{lim} \left(\frac{-\sin x}{2\sin x+x\cos x}\right) \\ Apply L\text{'}hospital rule \\ \log L=\frac{1}{2}\underset{x\to 0}{lim} \left(\frac{-\cos x}{3\cos x-x\sin x}\right) \\ \log L=\frac{{\displaystyle 1}}{{\displaystyle 2}}.\frac{{\displaystyle -1}}{{\displaystyle 3}} \\ L=e^{-1/6} \end{gathered}$$

Option(B)$\Rightarrow$$\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{\frac{1}{x^2}}$

$$\begin{gathered} L=\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{\frac{1}{x^2}} \\ Apply \log on both sides \\ \log L=\underset{x\to 0}{lim }\frac{\log \left(\frac{\tan x}{x}\right)}{{ }^{x^2}} \end{gathered}$$

Apply L'hospital rule

$$\begin{gathered} \log L=\underset{x\to 0}{lim}\frac{\left({\displaystyle \frac{x}{\tan x}\frac{xse{c}^{2}x-\tan x}{x^2}}\right)}{2x} \\ \log L=\underset{x\to 0}{lim}\left(\frac{xse{c}^{2}x-\tan x}{2x^2\tan x}\right) \end{gathered}$$

$$\begin{gathered} \log L=\frac{1}{2}\underset{x\to 0}{lim} \left[\frac{2xse{c}^{2}x \tan x}{2x\tan x+x^{2}se{c}^{2}x}\right] \\ \log L=\frac{1}{2}\underset{x\to 0}{lim} \left[\frac{2se{c}^{2}x \tan x}{2\tan x+xse{c}^{2}x}\right] \\ Numarator and Dinominator Divide by x then, \\ \log L=\frac{1}{3} \\ L=e^{\frac{1}{3}} \end{gathered}$$

Option(C)$\Rightarrow$$\underset{x\to 0}{lim}(\cos x{)}^{\frac{1}{\tan x}}$

$L=\underset{x\to 0}{lim}(\cos x{)}^{\frac{1}{\tan x}}$

$$\begin{gathered} apply \log on both sides \\ \log L=\underset{x\to 0}{lim}\frac{\log \left(\cos x\right)}{\tan x} \\ \log L=\underset{x\to 0}{lim}\frac{-\tan x}{se{c}^{2}x} \\ L=0 \end{gathered}$$

Option(D)$\Rightarrow$$\underset{x\to a}{lim}{(2-\frac{x}{a})}^{\mathrm{Tan}\left(\frac{\pi x}{2a}\right)}$

$L={\mathrm{e}}^{\underset{\mathrm{x}\to \mathrm{a}}{lim} \tan \left(\frac{\pi \mathrm{x}}{2\mathrm{a}}\right)\log \left(2-\frac{\mathrm{a}}{\mathrm{x}}\right)}$

$L={\mathrm{e}}^{\underset{\mathrm{x}\to \mathrm{a}}{lim} }\frac{\log \left(2-\frac{\mathrm{a}}{\mathrm{x}}\right)}{\cot \left(\frac{\pi \mathrm{x}}{2\mathrm{a}}\right)}$

$L=e^{\underset{x\to a}{lim}\frac{{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle \left(2-\frac{{\displaystyle a}}{{\displaystyle x}}\right)}}\frac{{\displaystyle a}}{{\displaystyle x^2}}}}{{\displaystyle -{\csc }^2\left(\frac{{\displaystyle \pi x}}{{\displaystyle 2a}}\right)\frac{{\displaystyle \pi }}{{\displaystyle 2a}}}} }$