Limit at Infinity Involving Number e

by Ifrah
(Somaliland)

Here we'll solve a limit at infinity submitted by Ifrah, that at first sight has nothing to do with number e. However, we'll use a technique that involves the limit deinition of e. The limit is:

Just to refresh your memory, the limit definition of e is:



In this case we use a simple change of variables. We force the limit to look like the limit definition of e:



Here, y is a variable we're now defining.

If we solve for x in the previous equation we get:



So, we note that x approaches infinity if and only if y also approaches infinity. So, replacing in the original limit we get:



Now, using the properties of exponents and limits we have:



Now we use the continuity of the power function and of exponents once again:



Now, the first factor becomes one, and the second factor is the definition of e! So:



And that's our answer.

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