$\text{ The values of }{a}_{w}b\text{ and }c\text{ such that }\underset{x\to 0}{lim} \frac{ax{e}^x-b\log (1+x)+cx{e}^{-x}}{x^2\sin x}=2$

$\begin{array}{l}\underset{x\to 0}{\lim }\frac{ax(1+x+\frac{x^2}{2!}+\mathrm{...})-b\left(x-\frac{x^2}{2}+\frac{x^3}{3}-\mathrm{...}\right)+cx\left(1-\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}\right)}{x^2\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}\right)}\\ \end{array}$

$\text{ Coefficient of }x\mathrm{\&}{x}^2=0\text{ and coefficient of }{x}^3=2$

$\Rightarrow a-b+c=0,a+\frac{b}{2}-c=0 and \frac{a}{2}-\frac{b}{3}+\frac{c}{2}=2$

Solving the above equations, we get a=3, b=12 and c=9