Question

If the following function $f\left(x\right)$ is continuous at $x=0$, find the values of $a,b$ and $c$.
$f\left(x\right)=\{\begin{array}{ccc}\frac{\sin (a+1)x+\sin x}{x},& if& x<0\\ c,& if& x=0\\ \frac{\sqrt{x+b{x}^2}-\sqrt{x}}{b{x}^{\frac{3}{2}}},& if& x>0\end{array}$.

Answer 1

If $f\left(x\right)$ is continuous at $x=0$, then

$f\left(0^{-}\right)=f\left(0\right)=f\left(0^{+}\right)$

$\Rightarrow f\left(0^{-}\right)={lim}_{x\to 0}(a+1)\frac{\sin (a+1)x}{(a+1)x}+{lim}_{x\to 0}\frac{\sin x}{x}$

$f\left(0^{-}\right)=a+2$

$f\left(0\right)=c$

$f\left(0^{+}\right)={lim}_{x\to 0}\frac{\sqrt{x}(\sqrt{1+bx}-1)}{bx\sqrt{x}}={lim}_{x\to 0}\frac{1+\frac{bx}{2}-1}{bx}$ (on expanding the square root)

$=\frac{1}{2}\therefore b\in R-\left\{0\right\}$

$c=2+a=\frac{1}{2}c=\frac{1}{2}$ $\Rightarrow a=\frac{1}{2}-2=\frac{-3}{2}$