Find $\lim\limits_{n\to\infty}{e^n – e^{\frac1n + n}}$

Question

So far I've tried: $${e^n – e^{\frac1n + n}} = e^n(1- e^{\frac1n}).$$ Then appling l'Hopitals rule to $$\lim\limits_{n\to\infty} e^n(1- e^{\frac1n}) = \lim\limits_{n\to\infty} \dfrac{(1- e^{\frac1n})}{e^{-n}},$$ I have not found success. Is there another way to manipulate the expression to be able to apply l'Hopitals? I know the limit should approach –$\infty$. To clarify, I can use l'Hopital.

Answer 1

HINT

We have that

$${e^n - e^{\frac1n + n}}=e^n \left(1-e^{\frac1n}\right)=-\frac{e^n}n \frac{e^{\frac1n}-1}{\frac1n}$$

then use standard limits.