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The following function was given to me $$f(x)=\lfloor x\rfloor+\lfloor-x\rfloor$$

wherein $\lfloor x\rfloor$ is the floor function of $x$. I was asked to select a proper sequence for showing that this function has no limit at $\infty$.

Honestly, my knowledge about analysis is weak. Thank you

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1 Answer 1

up vote 3 down vote accepted

Take $$f(n)=[n]+[-n]=n-n=0, \quad n\in\mathbb{N},$$ and then take $$f\left(\frac{2n+1}{2}\right)=n+(-n-1)=-1, \quad n\in\mathbb N$$

Then you have two different sequences with different limits when $\,n\to\infty\,$ and thus the limit at $\,\infty\,$ doesn't exist.

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Thank you Don, you saved me not to blush with my student. –  Babak S. Jun 24 '12 at 10:49
    
So, according to your selected sequences, can we say if Limit$_{x\rightarrow\infty}a_n=\infty$ and $\{f(a_n)\}$ is convergent, we have Limit$_{x\rightarrow\infty}f(x)$ undefined? –  Babak S. Jun 24 '12 at 11:02
    
@BabakSorouh Yes, see wikipedia –  Frank Science Jun 24 '12 at 14:11

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