does this limit diverge?

For clarification, little o notation usually means $f(n) = o(n)$ means $f$ grows slower than $n$, namely, $$ \lim_{n\to\infty} \frac{f(n)}{n} = 0 $$

To avoid confusion, let us consider $f_1(n), f_2(n)$ and $f_3(n)$ that are all $o(n)$. The original expression is reformulated by $$ \frac{n+f_1(n)}{\sqrt{n+f_2(n)}-\sqrt{n+f_3(n)}} = \frac{(n+f_1(n))(\sqrt{n+f_2(n)}+\sqrt{n+f_3(n)})}{(n+f_2(n))-(n+f_3(n))} $$ Note that the numerator is order of $n^{3/2}$ while the denominator is $f_2(n)-f_3(n) = o(n)$. Henceforth, the absolute value of the expression will diverge. If the sign of $f_2(n)-f_3(n)$ is known to be positive (negative) for all large enough $n$, then one can concludes that the limit is positive (negative) infinity. If the sign of $f_2(n)-f_3(n)$ switches infinitely often, then the limit does not exist.