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

Question

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

UpStudy AI · Accepted Answer

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:**

$$
\begin{array}{ccc}
f(x) = 5x^3 - x^4 \
\lim_{x \to \infty} f(x) = -\infty \
\lim_{x \to -\infty} f(x) = -\infty
\end{array}
$$
$$ \lim_{x \to \infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = -\infty $$

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