# Definition of exponential functions

• July 1st 2009, 01:27 AM
acc100jt
Definition of exponential functions
I do understand that $e$ is the number such that
$e=\lim_{n \rightarrow \infty} \left( 1+\frac{1}{n} \right)^n$
and how it is obtained.

But I don't understand how the definition of the function $e^x$ as a limit of a sequence is obtained, i.e.

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

Is there a proof?

Appreciate those who help. Thanks!
• July 1st 2009, 01:34 AM
simplependulum
We can expand it by Binomial Theorem

$( 1+ \frac{x}{n} )^n = 1 + (n) \frac{x}{n} + \frac{n(n-1)}{2} (\frac{x}{n})^2 + ....$

By taking the limit , n tends to infinity ,

$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + ....$
• July 1st 2009, 01:50 AM
acc100jt
great, thanks!