Give a limit expression that describes the left end behavior of the function. \( f(x)=\frac{5+2 x+9 x^{3}}{x^{3}} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \lim _{x \rightarrow-\infty} \frac{5+2 x+9 x^{3}}{x^{3}}=\square \) B. The limit does not exist and is neither \( -\infty \) nor \( \infty \).

To describe the left end behavior of the function \( f(x) = \frac{5 + 2x + 9x^{3}}{x^{3}} \), we evaluate the limit as \( x \) approaches negative infinity: \[ \lim_{x \to -\infty} \frac{5 + 2x + 9x^{3}}{x^{3}} \] Let's simplify the expression by dividing each term in the numerator by \( x^{3} \): \[ \lim_{x \to -\infty} \left( \frac{5}{x^{3}} + \frac{2x}{x^{3}} + \frac{9x^{3}}{x^{3}} \right) = \lim_{x \to -\infty} \left( \frac{5}{x^{3}} + \frac{2}{x^{2}} + 9 \right) \] As \( x \) approaches negative infinity: - \( \frac{5}{x^{3}} \) approaches 0. - \( \frac{2}{x^{2}} \) approaches 0. - \( 9 \) remains constant. Therefore, the limit is: \[ \lim_{x \to -\infty} \frac{5 + 2x + 9x^{3}}{x^{3}} = 9 \] **Correct Choice:** **A.** \( \lim_{x \rightarrow -\infty} \frac{5 + 2x + 9x^{3}}{x^{3}} = 9 \)