$\lim_{n \rightarrow \infty} n ((n^5 +5n^4)^{1/5} - (n^2 +2n)^{1/2})$ $$\lim_{n \rightarrow \infty} n ((n^5 +5n^4)^{1/5} - (n^2 +2n)^{1/2})$$
Please, help me to find the limit.

Factoring out an $n$ to make the expression $n^2((1+5/n)^{1/5}-(1+2/n)^{1/2})$ allows one to let $n=1/x$ and convert things to
$$\lim_{x\to0}{(1+5x)^{1/5}-(1+2x)^{1/2}\over x^2}$$
A couple of rounds of L'Hopital ought to finish things off.
