<s>[INST] Compute $ \sqrt{n^2 +3} - \sqrt{n^2 +2} $. I have a question concerning determining a basic limit of $ \sqrt{n^2 +3} - \sqrt{n^2 +2} $ as $n \rightarrow \infty $. Upon using the difference of squares, this yields: $ \frac{1}{\sqrt{n^2 +3} + \sqrt{n^2 +2}} $Now the limit seems intuitive, but I cant find a formal way of determining it. Any hints about what the next step should be would be appreciated.
 [/INST] Note that for any given $\epsilon>0$, we have 
$$\begin{align}
\left|\sqrt{n^2+3}-\sqrt{n^2+2}\right|&=\left|\frac{1}{\sqrt{n^2+3}+\sqrt{n^2+2}}\right|\\\\
&\le \frac{1}{2n}\\\\
&<\epsilon
\end{align}$$
whenever $n>N=1+\lfloor \frac2\epsilon\rfloor$.
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