Simple Limit

$\begin{aligned} L & = \lim_{x \to 0} \frac {e^x-2x-e^{-x}}{x-\sin x} & \small \color{#3D99F6} \text{By Maclaurin series} \\ & = \lim_{x \to 0} \frac {\left(1+x + \frac {x^2}{2!} + \frac {x^3}{3!} + \cdots \right)-2x-\left(1-x + \frac {x^2}{2!} - \frac {x^3}{3!} + \cdots \right)}{x-\left(x - \frac {x^3}{3!} + \frac {x^5}{5!} + \cdots \right)} \\ & = \lim_{x \to 0} \frac {2\left(\frac {x^3}{3!} + \frac {x^5}{5!} + \frac {x^7}{7!} + \cdots \right)}{\frac {x^3}{3!} + \frac {x^5}{5!} + \frac {x^7}{7!} + \cdots} \\ & = \boxed 2 \end{aligned}$

Reference: Maclaurin series