I was presented a question on

$\displaystyle \frac{cos(x)}{x}$ as x approaches infinity.

My approach was to separate the amplitude

My explination was since the amplitude was going to zero as x approached infinity, the limit was 0.

I was told this is not correct at all, which i fail to understand how my approach is not correct at all. In fact i can not imagine a single situation with cos(x) where if the amplitude is zero, the over all cos function would do anything but go to zero.

The problem, I was told, was that i did not use the squeeze theorem. i was expected to show

I do not understand how my approach was not correct at all vs using the squeeze theorem. Anyone explain where my logic if flawed?