Limit of g(x)sin(x) as x→0 where g is the Dirichlet function

[klarka]

Homework Statement

g(x) = 1 if x is rational, 0 if x is irrational.

Determine the limit, or prove it doesn't exist.

lim g(x)sinx
x->0

Homework Equations

The Attempt at a Solution

I said the limit is 0.

0 < |g(x)sinx| < |sinx|

lim |g(x)sinx|= 0
x->0
-|sinx| < g(x)sinx < sinx

lim g(x)sinx
x->0

The < are less than or equal. I can't figure out how to put them in correctly.
This was on my final and it's driving me crazy! I couldn't decided if the limit was 0 or didn't exist. If you could point out my mistakes I'd appreciate it!

 

[Dick]

No mistake. The limit is 0. -|sin(x)|<=g(x)*sin(x)<=|sin(x)|. The outer terms go to zero as z->0 so g(x)*sin(x) must also go to zero.