Proving Limit of (1+x/n)^n = exp(x)

[seboastien]

Maths project proving lim n-infinity (1+x/n)^n = exp(x)

Homework Statement

project aiming to show that limn→∞(1+x/n)^n = exp(x)

Homework Equations

The Attempt at a Solution

I have no Idea, is there any chance someone could give me some sort of clue?

 

[seboastien]

here is my attempt so far,
(a+b)^n= (n) a^n + (n) a^n-1 b + (n) a^n-2 b^2 + ... (n) b^n
(0) (1) (2) (n)

 

[seboastien]

oh please help me!
I don't know what to do!
should I use (a+b)(a+b)^n-1?

 

[bowma166]

Try rewriting your limit as $$\lim_{n\rightarrow\infty}\exp\left[\ln\left(\left(1+\frac{x}{n}\right)^{n}\right)\right]$$.

From there, I believe, you should try to get it into a form where L'Hôpital's rule will apply.

edit :: Yup, I just worked it out and this works.

 

[morphism]

How is exp(x) defined for you?

If it's defined as the inverse of the natural log, then bowma166's method is a good way to approach this problem. But be careful: first you must show that the limit

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

exists, for all x.