evaluate the limit as x approaches 0 from the right. of the function xcos(1/x)

Question

I know the answer is O but I am not sure as of my reasoning. I just that that since cos(1/x) is being multiplied by a number approching 0 that the whole limit would be o.

Answer 1

Let's just use an intuitive approach.
We know that if x -->0 , 1/x becomes infinitely large.
But cos(anything) is always a number between -1 and +1, no matter how large 1/x becomes
So we are then multiplying this by "almost zero" , so the answer will be zero.

Answer 2

more formally, for the purposes of homework, just use the comparison test

|cos(1/x)| <= 1, so

x cos(1/x) < x
which -> 0