(b) \( \lim _{x \rightarrow \infty} \frac{2 x^{3}+2 x-1}{x^{5}+2} \)

To find the limit, we can simplify the expression by dividing the numerator and the denominator by the highest power of \(x\) in the denominator, which is \(x^5\): \[ \lim_{x \rightarrow \infty} \frac{2 x^{3}+2 x-1}{x^{5}+2} = \lim_{x \rightarrow \infty} \frac{\frac{2 x^{3}}{x^{5}} + \frac{2 x}{x^{5}} - \frac{1}{x^{5}}}{\frac{x^{5}}{x^{5}} + \frac{2}{x^{5}}} \] This simplifies to: \[ \lim_{x \rightarrow \infty} \frac{\frac{2}{x^2} + \frac{2}{x^4} - \frac{1}{x^5}}{1 + \frac{2}{x^5}} \] As \(x\) approaches infinity, all terms containing \(x\) in the denominator approach zero. Therefore, we have: \[ \lim_{x \rightarrow \infty} \frac{0 + 0 - 0}{1 + 0} = \frac{0}{1} = 0 \] Thus, the final result of the limit is: \[ \boxed{0} \]