Evaluate the limit $\displaystyle\lim_{x \to 0}\frac{(e-\left(1 + x\right)^{1/x})}{\tan x}$. How to evaluate the following limit
$$\displaystyle\lim_{x\to 0} \dfrac{e-\left(1 + x\right)^{1/x}}{\tan x}$$
I have tried to solve it using L-Hospital's Rule, but it creates utter mess.
Thanks for your generous help in advance.

Since $e - (1+x)^{1/x} = \frac{xe}{2} + O(x)$ as $x \to 0,$ so
$$\frac{e - (1+x)^{1/x}}{\tan x} = \frac{\frac{xe}{2} + O(x)}{\sin x / \cos x} = \frac{\cos x \cdot [e/2 + O(x)]}{\sin x / x} \to \frac{e}{2}.$$
