Pliz help me with calculus limits proof of (e)

1. Oct 2, 2007

spektah

1. The problem statement, all variables and given/known data
Some on ehelp me prove this in detailed format using the knowledge of limits.

2. Relevant equations

lim(x-->infinity)(1+1/x)^x=e

Last edited: Oct 2, 2007
2. Oct 2, 2007

l46kok

Welcome to PF, spektah!

To receive help, you must either show your work or indicate on the part you are stuck on.

3. Oct 2, 2007

spektah

some body help.

4. Oct 2, 2007

spektah

i just need a starting point i dont know where to begin from?

5. Oct 2, 2007

malawi_glenn

according to my knowledge of calculus; that is one of the DEFINITIONS of e; you can't prove a definition.

But if you insert n = 1; you get 2.
And if you use the bionomial theorem, you can show that this limit is AT most 3.

So that 2< e < 3

So you can "only" show that this limit DOES exists.

http://en.wikipedia.org/wiki/Binomial_theorem

6. Oct 3, 2007

thanx a lot.

7. Oct 3, 2007

spektah

Help on partial derivatives...

Some one give me a starting point on this question
Givev all are partial derivatives.

Find (df/dx) and (df/dx) if f(x,y)=tan-1(y/x^(1/2))

8. Oct 3, 2007

Gib Z

i think you meant df/dx and df/dy, and when finding partial derivatives, treat every other variable as a constant except the one you are differentiating with respect to.

9. Oct 3, 2007

dynamicsolo

You can show that the limit is e, but I think you need something at least as strong as L'Hopital's Rule (after taking the natural logarithm of the expression and arranging the result into appropriate form) to prove it. I'm not aware of a nice shortcut.

10. Oct 3, 2007

malawi_glenn

According to my books, this is the definition of e, they (and me) could be wrong. I mean, the natural logarithm requires that you already have e and e^x right?

11. Oct 3, 2007

dynamicsolo

It is one definition of e, but it can be shown that this limit would have to have that value to be consistent with the definition of natural logarithm. I'd agree that there would be a certain circularity beyond that: you get as far as showing that this value must be the same as the base for natural logarithms.

(I had a look at what Wikipedia has on e, which jibes pretty much with the history I was familiar with. There are various ways to get e and, beyond that, you would just have checks for consistency.)

In any event, I think the OP was looking for a method of evaluating the limit, so they couldn't just say "it's the definition".

Last edited: Oct 3, 2007
12. Oct 3, 2007

malawi_glenn

The least thing we can say is that everything is coherent ;)