[Math] limit of absolute value of something equals absolute value of limit of something

Question

Does limit of absolute value of something always equal absolute value of limit of something?
Specifically, can I just say the below equality is true or do I need to prove it?
I'm not sure how to prove it.
Could you help me?

Answer 1

Hint: $f(x) = |x|$ is continuous.

(Of course, the limits must exist for this to apply. For a counterexample, see the comment above made by coffeemath).