# Use the ε-δ definition of a limit to prove this.

I know to how prove normal limits using the epsilon-delta definition, say:

limx→a f(x)=L But, there was a question on my textbook which I couldn't quite figure out to do, even though I've thought about it for a while I don't even know how to go about starting it.

Use ε-δ definition of a limit to prove that

limx→c f(x)=0 iff limx→c |f(x)|=0

Could anyone help me with this, even a hint on where to start? Thank you in advanc

• This is quite simple if you just start out by writing both limits in terms of $\epsilon$ and $\delta$. Have you done that? – user70962 Sep 25 '13 at 20:02

Hint: $|f(x)| = |(|f(x)|)|$, then applied the definition of limit
• (+1) It might even be clearer to say that $$|f(x)-0|=|f(x)|=|f(x)|-0=\bigl||f(x)|-0\bigr|.$$ – Cameron Buie Sep 25 '13 at 20:03