Problem: Prove that for all real numbers x, y we have
Solution: Consider the function
Thus we need to find out how large the fraction is, but
that expression is exactly like something from the
Mean value theorem. Are its
assumptions satisfied? If we take any two numbers
Thus we can estimate
This proves the claim.
Remark: If we put
This is nothing surprising for x large, since we know that arctangent is bounded and so x must sooner or later outgrow it, but it is interesting that this estimate also works for small x. Similar estimate can be shown (using analogous procedure) for the function sine.