SOLUTION: Use Calculus. For what value of a is the following equation true? lim ((x+a)/(x-a))^x = e x->inf

Question 1030353: Use Calculus. For what value of a is the following equation true?
lim ((x+a)/(x-a))^x = e
x->inf
Found 2 solutions by Edwin McCravy, robertb:
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!

That will be true if and only if the limit of the natural log is 1, so we reduce the problem to finding "a" if Since therefore And also Both numerator and denominator approach 0, so we can use L'Hopital's rule = 1 which simplifes to = 1 That limit is , so we have Edwin

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
We will use the definition of e to evaluate this:
Now let w = x-a.
==> x+a = w+2a, as , and

= .
In the very last expression, the second limit , because as w approaches infinity, and a is a fixed constant that doesn't vary with w.
But , and hence 2a = 1, or a = 1/2.
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