M L Aggarwal Solutions for Chapter: Continuity and Differentiability, Exercise 9: EXERCISE 5.9

Evaluate the limit (if exist):

$\underset{x\to \infty }{\lim }\frac{1}{{\left(x-3\right)}^2}$

Evaluate the limit (if exist):

$\underset{x\to \infty }{\lim }\frac{1}{{\left(1-x\right)}^2}$

Evaluate the limit (if exist):

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

Evaluate the limit (if exist):

$\underset{x\to 0}{\lim }\frac{7^x-1}{\tan x}$

Find the value of $k$, if

$\underset{x\to 0}{\lim }\frac{3^x-1}{\sqrt{2+x}-\sqrt{2}}=k\sqrt{2}\log 3$

Evaluate the limit (if exist):

$\underset{x\to 0}{\lim }\frac{e^x-x-1}{x}$

Evaluate the limit (if exist):

$\underset{x\to 1}{\lim }\frac{x-1}{\log x}$

Examine the function $f\left(x\right)=\left\{\begin{array}{l}{e}^{\frac{1}{x}} , x\ne 0\\ 1 , x=0\end{array}\right.$for continuity at $x=0.$