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

Use the general fact 
$$n(n+a-\sqrt[k]{n^k +akn^{k-1}})\rightarrow \frac{k+1}{2}a^2$$ as $n\to \infty$
to get a limit of
$$\frac{2+1}{2}-\frac{5+1}{2}=-\frac{3}{2}$$
For proof of the above fact.
If we let $A=\sqrt[k]{n^k +akn^{k-1}}$
then $$n((n+a)-A)=n\frac{(n+a)^k-A^k}{(n+a)^{k-1}+\cdots +A^{k-1}}
=\frac{\binom{k}{2}a^2n^{k-1}+\cdots }{(n+a)^{k-1}+\cdots +A^{k-1}}\rightarrow \frac{k+1}{2}a^2$$
