Question

$f\left(x\right)\{\begin{array}{c}\frac{e^x-1+ax}{x^2},x>0\\ bx=0\\ \frac{\sin \frac{x}{2}}{x}x<0\end{array}$ find $a+b$ if $f\left(x\right)$ is continuous at $x=0$.

Answer 1

Now,

$\begin{array}{c}{lim}_{x\to 0^{+}}\frac{e^x-1+a^x}{x^2}={lim}_{x\to 0^{+}}\frac{1+x+\frac{x^2}{2}-1+ax}{x^2}\\ \left(a+1\right)x=0=\frac{1}{2}\\ a=-1\\ b=\frac{1}{2}\\ {lim}_{x\to 0^{-}}\frac{\sin \frac{x}{2}}{x}{lim}_{x\to 0^{-}}\frac{\sin \frac{x}{2}}{2\times \frac{x}{2}}=\frac{1}{2}\\ a+b=-\frac{1}{2}\end{array}$