Mathematics - Limits Question with Solution | TestHub

${\mathrm{Lt}}_{x\to 0}\frac{a{e}^x-b\cos x+c.e^{-x}}{x\sin x}=2\Rightarrow a-b+c=0$

Apply LH Rule

$\underset{x\to 0}{\mathrm{Lt}}\frac{a{e}^x+b\sin x-c.e^{-x}}{\sin x+x\cos x}=2\Rightarrow a+0-c=0\Rightarrow a=c$

Apply LH rule

$\begin{array}{c}Lt\frac{a{e}^x+b\cos x+c{e}^{-x}}{\cos x+\cos x-x\sin x}=2\end{array}$

$\Rightarrow a+b+c=4$