\( 1< \) 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 focus on how the function behaves as \( x \) approaches infinity and negative infinity. First, let's calculate \( \lim_{x \rightarrow \infty} f(x) \): As \( x \) becomes very large, the term with the highest degree dominates the function. In this case, the term \( 4x^3 \) is the leading term. Therefore: \[ \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} (2x^2 + 4x^3 + 16) = \lim_{x \rightarrow \infty} 4x^3 = \infty \] Next, we find \( \lim_{x \rightarrow -\infty} f(x) \): Similarly, when \( x \) approaches negative infinity, the highest degree term will still dominate the function. Since \( 4x^3 \) will yield a large negative number, we get: \[ \lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow -\infty} (2x^2 + 4x^3 + 16) = \lim_{x \rightarrow -\infty} 4x^3 = -\infty \] So, to summarize: \[ \lim_{x \rightarrow \infty} f(x) = \infty \] \[ \lim_{x \rightarrow -\infty} f(x) = -\infty \]