calculus - How to evaluate $limlimits_{x rightarrow +infty}{e^x (e - (1+frac{1}{x} )^x)}$ without L'Hospital?

Using several times L'Hospital Rule I got $$\lim_{x \rightarrow +\infty}{e^x \left (e - \left(1+\dfrac{1}{x}\right )^x\right)} = +\infty.$$ Is it possible find this limit without L'Hospital?