Epsilon-delta proof of limit definition of e?

[ryoma]

Homework Statement

Prove that
$$\lim_{x\rightarrow\ 0} (1+x)^{1/x}=e$$
by an epsilon-delta proof.

Homework Equations

The Attempt at a Solution

I did:
x < a
1 + x < 1 + a
but I couldn't go any further.

 

[jgens]

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

 

[ryoma]

Anything other than the limit one.

 

[jgens]

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

 

[ryoma]

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

 

[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.

 

[ryoma]

Thank you. I understand it now.