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 .