R S Aggarwal Solutions for Exercise 1: EXERCISE

Find the value of $k$ for which

$f\left(x\right)=\left\{\begin{array}{c}\frac{\sin 2x}{5x},\text{ when }x\ne 0\\ k,\text{ when }x=0\end{array}\right.$ is continuous at $x=0$.

Find the value of $\lambda$ for which

$f\left(x\right)=\left\{\begin{array}{cc}\frac{x^2-2x-3}{x+1},& \text{ when }x\ne -1\\ \lambda ,& \text{ when }x=-1\end{array}\right.$ is continuous at $x=-1$.

For what value of $k$ is the following function continuous at $x=2$?

$f\left(x\right)=\left\{\begin{array}{r}2x+1,\text{ when }x<2\\ k,\text{ when }x=2\\ 3x-1,\text{ when }x>2\end{array}\right.$

For what value of $k$ is the following function

$f\left(x\right)=\left\{\begin{array}{c}\frac{x^2-9}{x-3},\text{ when }x\ne 3\\ k,\text{ when }x=3\end{array}\right.$ is continuous at $x=3$?

For what value of $k$ is the following function

$f\left(x\right)=\left\{\begin{array}{r}\frac{k\cos x}{\pi -2x},\text{ if }x\ne \frac{\pi }{2}\\ 3,\text{ if }x=\frac{\pi }{2}\end{array}\right.$ is continuous at $x=\frac{\pi }{2}$.

Find the values of $a$ and $b$ such that the following function is continuous. $f\left(x\right)=\left\{\begin{array}{l}5,\text{ when }x\le 2\\ ax+b,\text{ when }2<x<10\\ 21,\text{ when }x\ge 10\end{array}\right.$.

Find the values of $a$ for which the following functions $ƒ$, defined as $f\left(x\right)=\left\{\begin{array}{l}a\sin \frac{\pi }{2}(x+1),x\le 0\\ \frac{\tan x-\sin x}{x^3},x>0\end{array}\right.$ is continuous at $x=0$.

Prove that the function $f$ given by $f\left(x\right)=|x-3|, x\in R$ is continuous but not differentiable at $x=3$.