Explain how to find the value of each limit c) \( \lim _{x \rightarrow-\infty}-\frac{5 x^{5}+3 x^{3}+3}{6 x^{6}+2 x^{2}-9} \)

To find the limit as \( x \) approaches negative infinity for the expression \( -\frac{5 x^{5}+3 x^{3}+3}{6 x^{6}+2 x^{2}-9} \), we start by examining the degrees of the polynomials in the numerator and the denominator. The numerator has a degree of 5, while the denominator has a degree of 6. As \( x \) approaches negative infinity, the highest power terms dominate. To simplify, we divide both the numerator and the denominator by \( x^{6} \): \[ -\frac{\frac{5 x^{5}}{x^{6}}+\frac{3 x^{3}}{x^{6}}+\frac{3}{x^{6}}}{\frac{6 x^{6}}{x^{6}}+\frac{2 x^{2}}{x^{6}}-\frac{9}{x^{6}}} = -\frac{-5/x + 0 + 0}{6 + 0 - 0} = -\frac{-5/x}{6} \] As \( x \rightarrow -\infty \), \( -5/x \) approaches 0. Therefore, the limit evaluates to: \[ \lim_{x \to -\infty} -\frac{-5/x}{6} = -\frac{0}{6} = 0 \] Thus, the final answer is: \[ \lim _{x \rightarrow-\infty}-\frac{5 x^{5}+3 x^{3}+3}{6 x^{6}+2 x^{2}-9} = 0 \]