Mathematics - Limits Question with Solution | TestHub

We have

$$\begin{array}{rlrl}& \frac{{\lim }_{x\to 0}\frac{a{e}^x-b\cos x+c{e}^{-x}}{x\sin x}}{\left(1+x+\frac{x^2}{2!}+\dots \right)-b\left(1-\frac{x^2}{2!}+\dots \right)+c\left(1-x+\frac{x^2}{2!}+\dots \right)}& & \end{array}$$

$$=\underset{x\to 0}{\lim }\frac{a(x^2}{x^2}=\underset{x\to 0}{\lim }\frac{(a-b+c)+(a-c)x+x^2\left(\frac{a}{2!}+\frac{b}{2!}+\frac{c}{2!}\right)+0\left(x^3\right)}{x^2}$$

This limit exists if $a-b+c=0,a-c=0$ and is equal to 2 and if $(a+b+c)/2=2$, i.e., $a+b+c=4$. Solving these equations, we get $a=1,c=1$, and $b=2$

$$\frac{a}{b+c}=\frac{1}{3}=0.33$$