Limit at Infinity Involving Number e

by Ifrah

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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Nov 18, 2012
I like this method, one error in content
by: Anonymous

Right side should be:
lim(y -> inf):[1+(1/y)]^(5y+5)
instead of the 5y+y. This is evident in the next step, but confused me for a while.

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