Question

If $\displaystyle\lim _{x \rightarrow 0}\left(x^{-3} \sin 3 x+a x^{-2}+b\right)$ exists and is equal to zero, then the value of $a +2 b$, is $\ldots$

Answer 1

$\displaystyle\lim _{x \rightarrow 0} \frac{\sin 3 x}{x^{3}}+\frac{a}{x^{2}}+b$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\sin 3 x+a x+b x^{3}}{x^{3}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{3 \frac{\sin 3 x}{3 x^{3}}+a+b x^{2}}{x^{2}}$
for existence of limit $3 + a = 0$
$\Rightarrow a=-3$
$l=\displaystyle\lim _{x \rightarrow 0} \frac{\sin 3 x-3 x+b x^{3}}{x^{3}}$
$=27 \cdot \frac{\sin t-t}{t^{3}}+b=0$
$=-\frac{27}{6}+ b =0 $
$\Rightarrow b =\frac{9}{2}$
OR [Usign L’Hospital’s rule]
Hence, $a+2 b=-3+2 \times \frac{9}{2}=6$