Pliz help me with calculus limits proof of (e)

[spektah]

Homework Statement

Some on ehelp me prove this in detailed format using the knowledge of limits.

Homework Equations

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

 

[asd1249jf]

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[spektah]

some body help.

 

[spektah]

i just need a starting point i don't know where to begin from?

 

[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

 

[spektah]

thanx a lot.

 

[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))

 

[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.

 

[dynamicsolo]

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

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.

 

[malawi_glenn]

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.

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?

 

[dynamicsolo]

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?

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".

 

[malawi_glenn]

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