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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.

                       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
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Associated Topics:
High School Number Theory
High School Sequences, Series
High School Transcendental Numbers

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