*The following problem appeared in an examination of Calculus II and it is quite fun.*

**Problem**

- Give an example of a bounded function such that the limit does not exist. Give a brief explanation.
- Let be a function of the previous question . Examine whether the following limits exist:
Give a brief explanation.

**Solution**

- One such function , for example , is the following:
- For the limit we have that the limit is since
where is the bound of . Hence by squeezing the result follows. As of we have that the limit , in general , does not exist since we can write

and as we have seen in the first question the limit of as does not exist.