Question

Determine the values of $a,b,c$ for which
$\{\begin{array}{cc}f\left(x\right)=\frac{\sin (a+1)x+\sin x}{x}& \text{for}x<0\\ =c& \text{for}x=0\\ \frac{(x+b{x}^2{)}^{\frac{1}{2}}-x^{\frac{1}{2}}}{b{x}^{\frac{1}{2}}}& \text{for}x>0\end{array}$
is continuous at $x=0$.

Answer 1

$c=\underset{x\to 0^{-}}{lim}\frac{2\sin (a+1)x+\sin x}{x}=\underset{x\to 0^{-}}{lim}\frac{2\sin (\frac{a}{2}x+x)\cos \frac{a}{2}x}{x}=a+2$

$c=\underset{x\to 0^{+}}{lim}\frac{(x+b{x}^2{)}^{1/2}-x^{1/2}}{b{x}^{1/2}}=\underset{x\to 0^{-}}{lim}\frac{(1+bx{)}^{1/2}-1}{b}=0$

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

$⟹a+2=0⟹a=-2$