The Limit of (1+1/x)^x As x Approaches InfinityDate: 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/ |
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