5) $\lim _{x \rightarrow \infty} \sqrt{x^{2}+2 x}-\sqrt{x^{2}-8 x}$

Step 1 ：Given the expression \(\sqrt{x^{2}+2 x}-\sqrt{x^{2}-8 x}\), we are asked to find the limit as x approaches infinity.

Step 2 ：We can simplify the expression by multiplying and dividing by the conjugate. This will allow us to cancel out terms and simplify the expression to a form that we can easily evaluate the limit of.

Step 3 ：Let's rewrite the expression as \(-\sqrt{x(x - 8)} + \sqrt{x(x + 2)}\).

Step 4 ：By evaluating the limit of this expression as x approaches infinity, we find that the limit is 5.

Step 5 ：So, the final answer is \(\boxed{5}\).