real analysis - Proving limit relations with the exponential function

Prove the following limit relations:


$$\lim_{x\to0} (1+x)^{1/x} = e$$

$$\lim_{n\to\infty} \left(1 + \frac{x}{n}\right)^n = e^x$$

I'm not sure how to prove this as I'm not really sure what tools I have to prove it. I know by definition that the two limit relations are true, but any advice as to how to solve this specific problem/similar problems would be very appreciated!