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.

let $$y = (1 + x)^{1/x} ,\quad \ln y = \frac 1 x\ln(1 + x) = \frac 1x\left(x - \frac 12x^2 + \cdots\right) = 1 - \frac x 2 + \cdots $$ so that we have $$y = ee^{-x/2+\cdots} = e\left(1 - \frac x 2 + \cdots\right) \to e - y = \frac{ex}2+\cdots$$
therefore $$ \frac{e-y}{\tan x} = \frac{ex/2 + \cdots}{x + \cdots} \to \frac e2 \text{ as } x \to 0.$$
finally, we have $$\lim_{x \to 0}\frac{e-(1+x)^{1/x}}{\tan x} = \frac  e2.$$
