- #1

- 1,086

- 2

## Homework Statement

Find

[tex]\displaystyle\lim_{x\to\infty}x\sin(\frac{1}{x})[/tex]

where x is a natural number, and this is a sequence, not a real function.

## Homework Equations

## The Attempt at a Solution

I know the answer is 1, and that supposedly one solution is to introduce a dummy variable y = 1/x. However, I was wondering whether there is another way to solve the problem without doing that, perhaps by somehow applying the squeeze law. The best I can do is squeeze it between 0 and 1, but that's not good enough, of course. I've tried all sorts of trig identity rearrangements, and it gets me nowhere.

Also, if you do introduce that dummy variable, can you really just do that without having to explain why exactly the limit of the sequence (as x goes to infinity) is equal to the limit of the function you get, as y approaches 0? Or can you accommodate that by saying that a sequence is a function itself?

Thanks in advance.