Today we discuss the inferior limit of the product of two functions. Let us take the following simple question:
Question. Given a function with the following property
do we always have the following
It turns out that the statement should be hold since . Unfortunately, since the function blows up of order , the behavior of the product depends on the order of decay of the function . Let take the following counter-example.
We consider the function
Although the function is everywhere negative, there holds
Now it is clear that
By definition, one can easily prove that the statement of the question holds if we have