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