Question

If $f\left(x\right)=\frac{\sin 2x+a\sin x+b\cos x}{x^3}$ is continuous at $x=0$, find the values of $a$ and $b$.What is $f\left(0\right)?$

Answer 1

Using expansion,
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{(2x-\frac{{\left(2x\right)}^3}{3!}+...)+a(x-\frac{x^3}{3!}+...)+b(1-\frac{x^2}{2!}+...)}{x^3}$
$=\underset{x\to 0}{lim}\frac{(2+a)x-\frac{b{x}^2}{2}+x^3(-\frac{8}{6}-\frac{a}{6})+...}{x^3}=$finite
Above is posible only when $2+a=0$
and $b=0$
$\therefore a=-2,b=0$ and limit is
$-\frac{8}{6}-\frac{a}{6}=-\frac{8}{6}+\frac{2}{6}=-1.$
Since the function is to be continuous therefore Limit=value
$\therefore f\left(0\right)=-1.$