Continuity, proving that sin(x)sin(1/x) is continuous at 0.

[WaterPoloGoat]

Homework Statement

Define f(x)=sin(x)sin(1/x) if x does not =0, and 0 when x=0.

Have to prove that f(x) is continuous at 0.

Homework Equations

We can use the definition of continuity to prove this, I believe.

The Attempt at a Solution

I know from previous homework assignments that sin(x) is continuous at 0, but sin(1/x) is not. Is there a way I can use this knowledge to help me solve this problem, or do I need to start from scratch? If I need to start from scratch, do I apply the definition of continuity?

 

[phyzguy]

Apply the definition of continuity, and think about what these functions do as they approach zero. While sin(1/x) oscillates wildly, it remains bounded. So what is the limit of your function as x->0?