Calculate: $ \lim_{x \to 0 } = x \cdot \sin(\frac{1}{x}) $ Evaluate the limit:
$$ \lim_{x \to 0 } = x \cdot \sin\left(\frac{1}{x}\right) $$
So far I did: 
$$
\lim_{x \to 0 } = x\frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}\cdot x}
$$
$$
\lim_{x \to 0 } = 1 \cdot \frac{x}{x}
$$
$$
\lim_{x \to 0 } = 1
$$
Now of course I've looked around and I know I'm wrong, but I couldn't understand why. Can someone please show me how to evaluate this limit correctly? And tell me what I was doing was wrong.

You cant do this because
$$\sin(1/x)$$ isnt defined. So you cant use the fact that
$$\lim_{x\to 0} \sin(\alpha)/\alpha = 1$$
