Let $(a_n)_{n \geq 1}$ and $(b_n)_{n \geq 1}$ be two sequences of real numbers. If $b_n$ is positive, strictly increasing and unbounded and the following limit exists: $$ \lim_{n \rightarrow \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=l$$ Then the limit: $$\lim_{n \rightarrow \infty} \frac{a_n}{b_n}$$ also exists and it is equal to $l$ .

Remark. This theorem is also valid if $a_n$ and $b_n$ are monotone, tending to $0$ .