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!
If we let x=1 in the second limit is very easy to prove they are the same limit, you make a change of variable n=1/x, as x approaches 0, n tends to infinity, so you just replace in the limit and they both tend to e