# Epsilon-delta proof of limit definition of e?

1. Sep 16, 2010

### ryoma

1. The problem statement, all variables and given/known data
Prove that
$$\lim_{x\rightarrow\ 0} (1+x)^{1/x}=e$$
by an epsilon-delta proof.

2. Relevant equations

3. The attempt at a solution
I did:
x < a
1 + x < 1 + a
but I couldn't go any further.

2. Sep 16, 2010

### jgens

It would help to know what definition of e you're using.

3. Sep 16, 2010

### ryoma

Anything other than the limit one.

4. Sep 16, 2010

### jgens

Well, pick one that you like, and we'll work from there.

5. Sep 16, 2010

### ryoma

The only other one I really know is:
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$

6. Sep 17, 2010

### jgens

Okay, so we have our definition of e. Now, note that

$$\lim_{x\to 0}(1+x)^{1/x}=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}$$

Notice that you can expand the term on the right using the binomial theorem. If you can get this term to look like the one that you have for e, then it should be simple to show that their difference can be made as small as desired. Use this as the basis for your proof.

7. Sep 17, 2010

### ryoma

Thank you. I understand it now.