Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

enter image description here

share|cite|improve this question
If $a_n=(-1)^n$ then $\lim a_n$ doesn't exist (so neither does $|\lim a_n|$, yet $\lim |a_n|=1.$ – coffeemath Aug 9 '13 at 5:59
@AWertheim: Your proposed counterexample looks suspicious; you've brought a summation outside the absolute value, while the OP only brought out a limit. :) – Andrew D. Hwang Aug 9 '13 at 11:34
@AndrewD.Hwang goodness, how did I miss that? Tired eyes! Thanks. :) – Alex Wertheim Aug 9 '13 at 14:58
up vote 6 down vote accepted

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).

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.