\( 1< \) Describe the end behavior of the function \( f(x)=2 x^{2}+4 x^{3}+16 \) by finding \( \lim _{x \rightarrow \infty} f(x) \) and \( \lim _{x \rightarrow-\infty} f(x) \). \( \quad \begin{array}{l}\lim _{x \rightarrow \infty} f(x)=\square \text { (Simplify your answer.) } \\ \lim _{x \rightarrow-\infty} f(x)=\square \text { (Simplify your answer.) }\end{array} \)

To determine the end behavior of the function \( f(x) = 2x^{2} + 4x^{3} + 16 \), we analyze the limits as \( x \) approaches both positive and negative infinity. ### Step 1: Identify the Dominant Term The behavior of a polynomial function for large values of \( |x| \) is dominated by the term with the highest power of \( x \). In this case, the highest power is \( x^{3} \) with a coefficient of \( 4 \). ### Step 2: Determine the Limits 1. **As \( x \rightarrow \infty \):** \[ \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} (4x^{3} + 2x^{2} + 16) \] The \( 4x^{3} \) term dominates, and since \( x^{3} \) approaches \( +\infty \) as \( x \) does: \[ \lim_{x \rightarrow \infty} f(x) = \infty \] 2. **As \( x \rightarrow -\infty \):** \[ \lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow -\infty} (4x^{3} + 2x^{2} + 16) \] Again, the \( 4x^{3} \) term dominates. Since \( x^{3} \) approaches \( -\infty \) as \( x \) approaches \( -\infty \): \[ \lim_{x \rightarrow -\infty} f(x) = -\infty \] ### Conclusion \[ \begin{aligned} \lim_{x \rightarrow \infty} f(x) &= \infty \\ \lim_{x \rightarrow -\infty} f(x) &= -\infty \end{aligned} \]