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

$$\begin{array}{rlr}& \underset{x\to 0}{\mathrm{Lt}}\frac{a{e}^x+b\cos x+c{e}^{-x}}{\cos x+\cos x-x\sin x}=2\Rightarrow a+b+c=4& \therefore a+b+c=4\end{array}$$