Question

$f\left(x\right)=\begin{array}{c}=\frac{\sin ax}{2x}+b,ifx<0\\ =5,ifx=0\\ =2ax+b,ifx>0\end{array}$ , is continuous at $x=0$ ,find $a$ and $b$.

Answer 1

$\underset{x\to 0^{-}}{lim}f\left(x\right)=f\left(0\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)$

$\underset{x\to 0^{-}}{lim}\frac{sinax}{2x}+b=5$

$\Rightarrow \frac{a}{2}+b=5$

$\Rightarrow a+2b=10$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=f\left(0\right)$

$\underset{x\to 0^{+}}{lim}2ax+b=5$

$\Rightarrow b=5$

$a+10=10$

$\therefore a=0$

Hence, $a=0,b=5$