Proof of a limit as $x\to 0$

Question

I have been working on this one for some time now.

I was able to calculate the limit of $f(x)=|x|$ as $x\to 0$, yet I am unable to prove my result.

I tried using the $\varepsilon\text{-}\delta$ definition of a limit of a function, yet it just doesn't seem to work.

Answer 1

Let $\varepsilon>0. $ Then we can choose $\delta=\varepsilon$ such that $|x-0|<\delta$ implies $|f(x)-0|=||x|-0|<\delta=\varepsilon$ which proves the limit value of the given function at $0$ is $0.$