Question

If a function $f$ is continuous at $x=0$, where
$f\left(x\right)=\{\begin{array}{c}\frac{\sin 3x}{5x}+aforx<0\\ x+4-bforx\ge 0\end{array}$
find the value of $a+b$.

Answer 1

$\begin{array}{c}f\left(x\right)=\{\begin{array}{c}\frac{\sin 3x}{5x}+aforx<0\\ x+4-bforx\ge 0\end{array}\\ forcontinuity\\ \underset{x\to 0}{lim}\left(\frac{\sin 3x}{5x}\frac{3x}{3x}\right)+a=0+4-b\\ \frac{3}{5}+a=4-b\\ \Rightarrow a+b=4-\frac{3}{5}\\ \Rightarrow a+b=\frac{17}{5}\end{array}$