Proof of the Euler's constant limit

Question

I'm currently messing around with the limit of the Euler's constant. These two in particular: $$ \lim_{x\to 0} (1+x)^\frac{1}{x}=e $$ $$ \lim_{x\to \infty} (1+\frac{1}{x})^x=e $$ I really want to find an interpretation or a proof for them that doesn't use l'Hopital's rule, but I am quite lost to say the least. My current idea is to try finding a closed form for a finite sum $$ \sum_{x=0} ^{n} \frac{1}{x!} $$ and perhaps in the case where n tends to infinity it will match one of the limits. But I'm not too familiar with finding closed forms of series with factorials. I tried forming a reccurence relation and solving it, but it just loops back to a sum of factorials. Forming a differential equation doesn't seem to work either, because I have to differentiate a factorial or get zero, because n isn't in the formula for the general term and I just differentiate a constant. I'd really appriciate any help with this! And terribly sorry if this is actually simple and I'm just overlooking something.

Answer 1

Take a look at a Riemann sum of size one under $1/x$.

Get the double inequality $$1/(n+1)\le\int_n^{n+1}1/x\rm dx \le1/n$$

Which leads to $$1/(n+1)\le \ln((n+1)/n)\le1/n$$

Now take $e$ $$e^{1/(n+1)}\le 1+1/n\le e^{1/n}$$.

Raise to the $n$, and you can take limits

$$e^{n/(n+1)}\le (1+1/n)^n\le e$$.

A little adjustment will get you $e^x=\lim_{n\to\infty}(1+x/n)^n$.