Determine the limit of the function as x approaches infinity:
$\lim _{x \rightarrow \infty} \frac{8 x^{2}-x+10}{\sqrt{x^{2}+2 x-8}}$

Step 1 ：Determine the limit of the function as x approaches infinity: \(\lim _{x \rightarrow \infty} \frac{8 x^{2}-x+10}{\sqrt{x^{2}+2 x-8}}\)

Step 2 ：The function is a rational function where the degree of the numerator is greater than the degree of the denominator. In such cases, the limit as x approaches infinity is determined by the highest degree term in the numerator and the denominator.

Step 3 ：Here, the highest degree term in the numerator is \(8x^2\) and in the denominator is \(\sqrt{x^2}\). We can divide all terms by \(x^2\) to simplify the function and then find the limit.

Step 4 ：\(f = \frac{8*x^{2} - x + 10}{\sqrt{x^{2} + 2*x - 8}}\)

Step 5 ：\(f_{simplified} = \frac{8*x^{2} - x + 10}{x^{2}\sqrt{x^{2} + 2*x - 8}}\)

Step 6 ：The limit of the function as x approaches infinity is 0. This is because the highest degree term in the numerator is \(8x^2\) and in the denominator is \(x^2\) (after simplification). As x approaches infinity, the \(x^2\) terms dominate and the limit is determined by the ratio of the coefficients of these terms, which is 0.

Step 7 ：Final Answer: \(\boxed{0}\)