Take the 2-minute tour ×
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.

I think the limit is $1$, because $\left\lceil \frac{1}{n}\right\rceil=1$ for all $n$, but it seems counter intuitive for some reason. If we crudely "replace $n$ with $\infty$" we get $\lceil 0\rceil=0$ (also, if it was $0$ I can't see a way of proving it).

Is this due to the discontinuous character of $\lceil x\rceil?$

share|improve this question
3  
You’re right, and it shows the undependibility of intuition; or alternatively, that the discontinuous functions are the ones that go against intuition. –  Lubin Mar 4 '13 at 15:27

1 Answer 1

up vote 12 down vote accepted

You are exactly correct - the limit is $1$, because the sequence is constant. The limit of $\left\lceil-\frac{1}{n}\right\rceil$ as $n\to\infty$ is $0$, and this demonstrates the discontinuity of $\lceil x\rceil$.

If the limit were $0$, the sequence would need to become arbitrarily small as $n$ increased. So for example, there should be some $n$ such that $\left\lceil\frac{1}{n}\right\rceil<\frac{1}{2}$, but there isn't, because $\left\lceil\frac{1}{n}\right\rceil=1$ for all $n$.

share|improve this answer

Your Answer

 
discard

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.