Associated Topics || Dr. Math Home || Search Dr. Math

### The Limit of (1+1/x)^x As x Approaches Infinity

```
Date: 02/17/98 at 12:46:06
From: Heyward Harvey
Subject: the limit of (1+1/x)^x as x approaches infinity

Dear Dr. Math,

I am a senior at Ashley Hall School in Charleston, South Carolina. In
my AP Calculus AB class we recently began studying e. I was wondering
how Euler found the numerical value of e. I also would like to know
what the significance of the equation (1+1/x)^x is, and how they found
out that the value of the limit as it approaches infinity is e.

Sincerely,
Heyward Harvey
```

```
Date: 02/17/98 at 16:55:28
From: Doctor Rob
Subject: Re: the limit of (1+1/x)^x as x approaches infinity

Euler used the infinite series

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ...

to compute e to the accuracy he needed.

The expression (1+1/x)^x occurs when you try to take the derivative of
log_a(t), for a constant a, with respect to t. The difference quotient
can be rewritten to involve this expression, and you want to take the
limit as x -> infinity.

First you prove that this is an increasing function of x, and that it
is bounded above by some constant. That proves that the limit exists.
At first you don't know the value of this limit, but you can
approximate it by computing this expression for a large value of x.
Later, you find the series above for e, which allows you to calculate
its value to as much accuracy as desired.

-Doctor Rob, The Math Forum
Check out our web site http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
High School Sequences, Series
High School Transcendental Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/