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# On a bounded function and its limits

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

Problem

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

Give a brief explanation.

Solution

1. One such function , for example , is the following:

2. 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.