<s>[INST] Limit of this sequence $\lim_{n\to \infty}\frac{ \sqrt{2n+2} - \sqrt{2n-2}}{\sqrt{3n+1} - \sqrt{3n}}$ I am trying to calculate the limit of this sequence :
$$\lim_{n\to \infty}\frac{ \sqrt{2n+2} - \sqrt{2n-2}}{\sqrt{3n+1} - \sqrt{3n}}$$
I tried two methods and the two methods leaded me to infinity or 4/0.
Anything would be helpful , thanks.
 [/INST] Call your limit $L$. Since $\sqrt{a+b}-\sqrt{a}=\frac{b}{\sqrt{a+b}+\sqrt{a}}$, $$L=\lim_{n\to\infty}\frac{\sqrt{2n+2}-\sqrt{2n-2}}{\sqrt{3n+1}-\sqrt{3n}}=\lim_{n\to\infty}\frac{4(\sqrt{3n+1}+\sqrt{3n})}{\sqrt{2n+2}+\sqrt{2n-2}}.$$Since $\sqrt{an+b}\sim\sqrt{an}$ for large $n>0$ and $a>0$,$$L=\lim_{n\to\infty}\frac{8\sqrt{3n}}{2\sqrt{2n}}=\frac{8\sqrt{3}}{2\sqrt{2}}=2\sqrt{6}.$$
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