<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] We can use that
$$ \frac{ \sqrt{2n+2} - \sqrt{2n-2}}{\sqrt{3n+1} - \sqrt{3n}}=$$
$$=\frac{ \sqrt{2n+2} - \sqrt{2n-2}}{\sqrt{3n+1} - \sqrt{3n}}
\cdot \frac{\sqrt{2n+2} + \sqrt{2n-2}}{ \sqrt{2n+2} + \sqrt{2n-2}}
\cdot\frac{\sqrt{3n+1} + \sqrt{3n}}{ \sqrt{3n+1} + \sqrt{3n}}=$$
$$=4\cdot \frac{\sqrt{3n+1} + \sqrt{3n}}{ \sqrt{2n+2} + \sqrt{2n-2}}=4\cdot \frac{\sqrt{3+\frac1n} + \sqrt{3}}{ \sqrt{2+\frac2n} + \sqrt{2-\frac2n}}\to 4\frac{2\sqrt 3}{2\sqrt 2}=2\sqrt 6$$
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