Let and be functions, both differentiable in a neighborhood of 0, with and such that

Prove or disprove each the following statements.

- as .
- as .

- True.

*Proof.*Since as we know by the definition of thatThus, for every there exists a such that

So, for we have

The final line follows since by hypothesis. Therefore,

Hence,

By definition, we then have

- False.

Consider for and for . Then, for ,For we have .

Next,

Since we have as . However, since

does not exist.

There is a fault in your last step. lim(x->0) sin(1/x)/(1/x) = 0 instead of 1.