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):
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
= .
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.
