\( \left. \begin{array} { c } { f ( x ) = 5 x ^ { 3 } - x ^ { 4 } } \\ { \lim _ { x \rightarrow \infty } f ( x ) = \square } \\ { \lim _ { x \rightarrow - \infty } f ( x ) = \square } \end{array} \right. \)

Let's analyze the function \( f(x) = 5x^3 - x^4 \) to determine the limits as \( x \) approaches both \( \infty \) and \( -\infty \). ### 1. Limit as \( x \to \infty \) \[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} (5x^3 - x^4) \] - **Dominant Term:** As \( x \) becomes very large, the term with the highest power of \( x \) dominates. In this case, \( -x^4 \) dominates over \( 5x^3 \). - **Behavior:** The term \( -x^4 \) becomes increasingly negative as \( x \) increases. \[ \lim_{x \to \infty} f(x) = -\infty \] ### 2. Limit as \( x \to -\infty \) \[ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (5x^3 - x^4) \] - **Dominant Term:** Again, \( -x^4 \) is the dominant term. - **Behavior of Each Term:** - \( x^4 \) is always positive, so \( -x^4 \) is always negative, regardless of whether \( x \) is positive or negative. - \( 5x^3 \) becomes negative as \( x \) approaches \( -\infty \) because \( x^3 \) retains the sign of \( x \). - **Combined Behavior:** Both terms become large in magnitude, but \( -x^4 \) grows much faster than \( 5x^3 \). \[ \lim_{x \to -\infty} f(x) = -\infty \] ### Summary \[ \begin{array}{ccc} f(x) = 5x^3 - x^4 \\ \lim_{x \to \infty} f(x) = -\infty \\ \lim_{x \to -\infty} f(x) = -\infty \end{array} \] **Answer:** \[ \begin{array}{ccc} f(x) = 5x^3 - x^4 \\ \lim_{x \to \infty} f(x) = -\infty \\ \lim_{x \to -\infty} f(x) = -\infty \end{array} \]