Is the Function f(x) Continuous at x=0 Given Its Definition and Behavior?

[kidia]

Using the inequalities -1 = < sin 1/x = < 1 for x not equal to 0 determine wether or not the function
f(x)= {sinxsin1/x x is not equal to 0
f(x)= {0 x=0

continuous at x=0

 

[Moo Of Doom]

Hmm...

Well, we obviously must find whether or not
$$\lim_{x\rightarrow0}\sin{x}\sin{\frac{1}{x}}=0$$

Which is a bit of a challenge since $$\lim_{x\rightarrow\infty}\sin{\frac{1}{x}}$$ is undefined.
Looks like squeeze theorem time.
For all x, $$-|x|\leq\sin{x}\leq|x|$$ and $$|\sin{\frac{1}{x}}|\leq1$$.
Therefore, $$-|x|\leq\sin{x}\sin{\frac{1}{x}}\leq|x|$$ for all x.
Since $$\lim_{x\rightarrow0}-|x|=\lim_{x\rightarrow0}|x|=0, \lim_{x\rightarrow0}\sin{x}\sin{\frac{1}{x}}=0$$

And thus it is continuous