Problem
\( \lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+5 n}-\sqrt{n^{2}+n}\right) \)
Solution
Step 1 ：\(\lim_{n\rightarrow\infty} \frac{(n^2+5n)-(n^2+n)}{\sqrt{n^2+5n}+\sqrt{n^2+n}}\)
Step 2 ：\(\lim_{n\rightarrow\infty} \frac{4n}{\sqrt{n^2+5n}+\sqrt{n^2+n}}\)
Step 3 ：\(\lim_{n\rightarrow\infty} \frac{4}{\frac{\sqrt{n^2+5n}}{n}+\frac{\sqrt{n^2+n}}{n}}\)
