Question

If function $f\left(x\right)$ is continuous in interval $[-2,2]$, find the value of $(a+b)$ where $f\left(x\right)=\{\begin{array}{c}\frac{\sin ax}{x}-2,for-2\le x<0\\ 2x+1,for0\le x\le 1\\ 2b\sqrt{x^2+3}-1,for1<x\le 2\end{array}$

Answer 1

f(x) is continuous at x=0

$\begin{array}{c}LHS=RHLatx=0\\ {lim}_{x\to 0}\frac{\sin ax}{x}-2={lim}_{x\to 0}2x+1\\ {lim}_{x\to 0}\frac{a\sin ax}{ax}-2=1\\ a-2=1\\ a=3\end{array}$

f(x) is continuous at x=1

$\begin{array}{c}LHL=RHLatx=1\\ {lim}_{x\to 1}2x+1={lim}_{x\to 1}2b\sqrt{x^2+3}-1\\ 3=2b\times -1\\ b=1\\ a+b=3+1=4\end{array}$