limit of absolute value of something equals absolute value of limit of something?

real-analysis calculus

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?

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Solution 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).

Solution 2:

Assuming both of your limits exist, then the absolute value of the limit will be the limit of the absolute value, by continuity of absolute value. However, it's entirely possible that taking the absolute value will cause a limit to exist where it hadn't before.

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