Just for practice!#2

$\large{\frac{x(ae^x+ce^{-x})-b\ln (1+x)}{x^2 \sin x}}$

using series expansion of $e^{x},e^{-x},\sin x ,\ln (1+x)$

$\large{\frac{x(a+ax+\frac{ax^2}{2}+..+c-cx+\frac{cx^2}{2}-...)-b(x-\frac{x^2}{2}+\frac{x^{3}}{3}-...)}{x^2 (x-\frac{x^{3}}{3!}+...)}}$

collecting terms of $x,x^{2}~and ~x^3$

$\large{\frac{x(a-b+c)+x^{2}(a-c+\frac{b}{2})+x^{3}(\frac{a}{2}+\frac{c}{2}-\frac{b}{3})}{x^{3} (1-\frac{x^2}{3!}+..)}}$

For limit to exist

$\large{\begin{cases} a-b+c=0 \\ a-c+\frac { b }{ 2 } =0 \\ \frac { a }{ 2 } +\frac { c }{ 2 } -\frac { b }{ 3 } =2 \end{cases}}$

solving we get $\boxed{a=9}\quad \boxed{b=12}\quad \boxed{c=3}$