Question: Consider the following rational function $f$.
$f(x) = \frac{-6x^5 + 8x^4 + 3x}{6x^2 - 4x - 5}$
Determine $f$'s end behavior.
$f(x) \to$ pick value as $x \to -\infty$.
$f(x) \to$ pick value as $x \to \infty$.
1. **State the problem:** We want to find the end behavior of the rational function
$$f(x) = \frac{-6x^5 + 8x^4 + 3x}{6x^2 - 4x - 5}.$$
This means we want to find the limits of $f(x)$ as $x \to -\infty$ and as $x \to \infty$.
2. **Recall the rule for end behavior of rational functions:**
The end behavior is dominated by the terms with the highest powers of $x$ in the numerator and denominator.
3. **Identify the leading terms:**
- Numerator leading term: $-6x^5$
- Denominator leading term: $6x^2$
4. **Simplify the function by focusing on leading terms:**
$$f(x) \approx \frac{-6x^5}{6x^2} = \frac{\cancel{-6}x^{5}}{\cancel{6}x^{2}} = -x^{3}.$$
5. **Analyze the limit as $x \to \infty$:
Since $f(x) \approx -x^3$, and as $x \to \infty$, $-x^3 \to -\infty$.
6. **Analyze the limit as $x \to -\infty$:
As $x \to -\infty$, $-x^3 = - (\text{negative large number})^3 = - (\text{negative large number}) = \text{positive large number}$.
So $f(x) \to \infty$ as $x \to -\infty$.
**Final answers:**
- $\lim_{x \to -\infty} f(x) = \infty$
- $\lim_{x \to \infty} f(x) = -\infty$