Math Help - Epsilon Delta Limit Proof sinxcos(1/x)

1. Epsilon Delta Limit Proof sinxcos(1/x)

I'm completely stuck on the limit of sinxcos(1/x) as x goes to 0.

If you are allowed to take the arcsin of epsilon, then I should be fine, but I have a feeling you aren't allowed to do that. Any hints?

2. Re: Epsilon Delta Limit Proof sinxcos(1/x)

Originally Posted by jsmith90210
I'm completely stuck on the limit of sinxcos(1/x) as x goes to 0.
Surely you can see that $\left| {\sin (x)\cos \left( {\frac{1}{x}} \right)} \right| \le \left| {\sin (x)} \right|.$

3. Re: Epsilon Delta Limit Proof sinxcos(1/x)

yes, I can see that, but then I have abs(sinx)<epsilon, i need to get from there to abs(x)<some variation of epsilon and I don't know how to do that

4. Re: Epsilon Delta Limit Proof sinxcos(1/x)

Originally Posted by jsmith90210
yes, I can see that, but then I have abs(sinx)<epsilon, i need to get from there to abs(x)<some variation of epsilon and I don't know how to do that
Do you know that $|\sin(x)|\le |x|~?$