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Let $(a_n)_n, (b_n)_n$ be two real sequences s.t. $(b_n)_n$ is strictly increasing and unbounded. Prove that $$\lim_{n\rightarrow \infty} \frac {a_n}{b_n}= \lim_{n\rightarrow \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}$$ if limit on the right exists.

I could prove this. But I thinking whether the converse is true, namely: if the limit on the left exists, what about the right limit? Does it exist? Or are there counterexamples?

EDIT: There is this post that answers my question, but does not have much content.

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Let $a_n=\sqrt n +(-1)^n$ and $b_n=\sqrt n$. Then the limit on the left is $1$, whereas the general term on the right is $$ \begin{align}\frac{\sqrt{n+1}+(-1)^{n+1}-\sqrt n-(-1)^n}{\sqrt{n+1}-\sqrt n}&=1-(-1)^n\frac{2}{\sqrt{n+1}-\sqrt n}\\&=1-2(-1)^n(\sqrt{n+1}+\sqrt n)\\&\approx 1\pm4\sqrt n\end{align}$$ and no limit exists on the right.

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  • $\begingroup$ Wow. Could you give some rough idea on how did you think through this problem? $\endgroup$ – math Oct 14 '13 at 21:13

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