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June 13th, 2019, 07:08 PM  #1 
Senior Member Joined: Oct 2013 From: Far far away Posts: 429 Thanks: 18  The mathematical constant e (Euler's constant)
The mathematical constant e (2.7182818...) is given by the formula: e = limit of (1 + 1/x)^x as x approaches infinity. I used the graphing tool Desmos and graphed the function y = (1 + 1/x)^x and saw the following results: 1. Horizontal asymptote at y = e 2. vertical asymptote at x = 1 3. a gap/break in the function at 1 < x < 0. This is the region where the function is undefined as per the graph. MY QUESTIONS: 1. I can understand that as x approaches + infinity, the limit of the function approaches e or 2.71828... but what I can't understand is why the function approaches the same limit (e) when x approaches negative infinity: suppose we confine ourselves to the negative part of the number line. If we do this we get x as x. Substituting this value into the function we get: (1 + 1/(x))^x = ((x + 1)/x)^x = ((x  1)/x)^x = ((x  1)/x)^x = (x/(x  1))^x or $\displaystyle \left(\frac{x}{x1}\right)^{x}$ The original function, with a little manipulation, for e = $\displaystyle \left(\frac{x+1}{x}\right)^{x}$ $\displaystyle \left(\frac{x}{x1}\right)^{x}$ is NOT the same as $\displaystyle \left(\frac{x+1}{x}\right)^{x}$ and yet both tend to e as x approaches either positive or negative infinity. I tried to reason it like this: Let's compare $\displaystyle \left(\frac{x}{x1}\right)^{x}$ with $\displaystyle \left(\frac{x+1}{x}\right)^{x}$ I tried to find the ration between these two expressions and got the following: $\displaystyle \frac{\left(\frac{x}{x1}\right)^{x}}{\left(\frac{x+1}{x}\right)^{x}}= \left(\frac{x^{2}1}{x^{2}}\right)^{x} = z$ Now z approaches 1 as x tends to either + or  infinity. In other words the two expressions $\displaystyle \left(\frac{x}{x1}\right)^{x}$ and $\displaystyle \left(\frac{x+1}{x}\right)^{x}$ CONVERGE at a number which here is e or 2.71828... Is my reasoning correct? 2. How do you explain the gap/break in the function at 1<x<0. I tried a few examples and found out that the value of the function $\displaystyle \left(\frac{x+1}{x}\right)^{x}$ is sometimes + and sometimes negative. Is this the reason why the graph isn't plotted in the interval 1<x<0? Also some times the function in this interval evaluates to an imaginary number. Any help will be deeply appreciated. Thanks 
June 13th, 2019, 07:34 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
Define two functions \[f(x) = \left(\frac{x}{x1}\right)^x \quad g(x) = \left(\frac{x+1}{x}\right)^x \] and notice that \[f(x+1) = \left(\frac{x+1}{x}\right)^{x+1} = \left(\frac{x+1}{x}\right)g(x) .\] Now $f$ and $g$ are both continuous for $x > 0$ and so \[\lim\limits_{x \to \infty} f(x) = \lim\limits_{x \to \infty} f(x+1) = \lim\limits_{x \to \infty} \left(\frac{x+1}{x}\right) \cdot \lim\limits_{x \to \infty} g(x) = 1 \cdot e \] 
June 13th, 2019, 08:07 PM  #3  
Senior Member Joined: Oct 2013 From: Far far away Posts: 429 Thanks: 18  Quote:
 

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