calculus - How do I solve $lim_{x to infty} x(e^{(1/x)}-1)$ without L'Hopital?

I don't have experience with L'Hopital Rule nor series and thats what most solution are, is there is other method can be used to solve that limit? i thought about trying to use first principle of derivative but i dont know where to begin. i need some help to guide me to the right direction.

Answer

This is precisely

$$\lim_{x \to +\infty} \frac{\exp\left(\frac 1x\right) - 1}{\frac 1x} = \lim_{u \to 0^{+}} \frac{\exp\left(u \right) - 1}{u}$$

Does that remind you of some derivative?