\begin{tabular}{|l} I can determine the enc \\ Determine the end behavior of \( f(x)=2 x^{9}-3 x^{4}+x^{3}+2 x-7 \) \\ \( \lim _{x \rightarrow-\infty} f(x)= \) \\ \( \lim _{x \rightarrow \infty} f(x)= \) \\ \end{tabular}

To determine the end behavior of the polynomial \( f(x) = 2x^9 - 3x^4 + x^3 + 2x - 7 \), we look at the leading term, which is \( 2x^9 \). As \( x \) approaches positive infinity (\( x \rightarrow \infty \)), the term \( 2x^9 \) dominates, and since it is a positive coefficient and an odd power, the limit goes to positive infinity. Thus, \( \lim_{x \rightarrow \infty} f(x) = \infty \). As \( x \) approaches negative infinity (\( x \rightarrow -\infty \)), the leading term \( 2x^9 \) still dictates the behavior. An odd power with a positive coefficient will yield negative infinity. Therefore, \( \lim_{x \rightarrow -\infty} f(x) = -\infty \). In summary: \[ \lim _{x \rightarrow-\infty} f(x) = -\infty \] \[ \lim _{x \rightarrow \infty} f(x) = \infty \]