Limit infinite

Like so:

$(\sqrt{n^3+7n^2}-\sqrt{n^3})/\sqrt n \rightarrow \sqrt{n^2+7n}-\sqrt{n^2}\\ \rightarrow (n^2+7n)^{1/2}-n \rightarrow n(1+7/n)^{1/2}-n \\ \rightarrow n(1+(1/2)(7/n)+(1/2)(-1/2)(1/2!)(7/n)^2+...)-n\\ \rightarrow n + 7/2 - 49/(8n) + ... - n \rightarrow 7/2 - 49/(8n) + ...$

As all the higher order terms involve a power of n in the denominator, they go to zero as n goes to infinity, so we are left with just 7/2 as the limit.