Today, I have been asked to calculate the following limit

for each fixed . From the mathematical point of view, we can assume as we just replace by if necessary.

There are three possible cases

**Case 1**. . In this case, it is well known that function is monotone decreasing since

in its domain. Consequently, it holds

It turns out that

Keep in mind that

as . Thus

**Case 2**. . Due to the fact that is odd with respect to , we can replace by to reach to the result

**Case 3**. . This is trivial.

Taking into account above cases we deduce that

for any .

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Hi! Here’s more simple proof.

Set , . The claim is that that the limit exists and is zero. To prove it, we may assume that . Since , is a decreasing sequence of nonnegative numbers, hence has a limit. The limit, L, must be zero, which follows from .

Comment by Volodja — January 11, 2011 @ 3:23

Oh yes, that was the traditional way we all know from the high school. Thanks a lot guy.

Comment by Ngô Quốc Anh — January 11, 2011 @ 11:46