The problem asks which option describes the end behavior of a polynomial function, using limit notation as $x \to \infty$ and $x \to -\infty$.

Let's analyze each option one by one, based on the typical behavior of polynomial functions.

Option 1:

$\lim_{x \to \infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty$ This describes a polynomial with a negative leading coefficient and odd degree. For such functions:

• As $x \to \infty$, the polynomial tends toward $-\infty$.
• As $x \to -\infty$, the polynomial tends toward $\infty$.

Option 2:

$\lim_{x \to \infty} f(x) = 8 \quad \text{and} \quad \lim_{x \to -\infty} f(x) = -8$ This behavior is inconsistent with a polynomial function. Polynomials grow without bound as $x \to \infty$ or $x \to -\infty$, so having limits that approach constants (like 8 and -8) suggests this isn't the correct option.

Option 3:

$\lim_{x \to \pm \infty} f(x) = 1$ This suggests the function tends toward a constant as $x$ approaches both positive and negative infinity, which is not possible for a polynomial function.

Option 4:

$\lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = -\infty$ This describes a polynomial with a positive leading coefficient and odd degree. For such functions:

• As $x \to \infty$, the polynomial tends toward $\infty$.
• As $x \to -\infty$, the polynomial tends toward $-\infty$.

Conclusion:

The correct answer is Option 4, which describes the behavior of a polynomial with a positive leading coefficient and odd degree.

Do you have any questions or would you like more details?

Here are some related questions to expand on this topic:

1. What determines the end behavior of a polynomial function?
2. How can you determine the degree of a polynomial from its end behavior?
3. What is the role of the leading coefficient in determining the end behavior of polynomials?
4. How does the end behavior of even-degree polynomials differ from odd-degree polynomials?
5. How do polynomial functions differ from other functions, such as exponential functions, in terms of end behavior?

Tip: Always check the leading term of a polynomial to determine its end behavior, since the highest degree term dominates as $x \to \infty$ or $x \to -\infty$.