Let's analyze the given functions and describe their end behavior using limit notation.

Problem 14: $g(x) = 4x^7 - 13x - 2$

End Behavior

For large positive or large negative values of $x$, the term with the highest degree, $4x^7$, dominates the behavior of the function.

• As $x \to \infty$ (as $x$ approaches positive infinity), the dominant term is $4x^7$, which grows without bound. So, $\lim_{x \to \infty} g(x) = \infty$

• As $x \to -\infty$ (as $x$ approaches negative infinity), the term $4x^7$ becomes negative because $x^7$ is negative for negative $x$. Therefore, $\lim_{x \to -\infty} g(x) = -\infty$

Limit notation:

$\lim_{x \to \infty} g(x) = \infty, \quad \lim_{x \to -\infty} g(x) = -\infty$

Problem 15: $f(x) = 5x^{14} - 4x^5 - 6x + 8$

End Behavior

The term with the highest degree, $5x^{14}$, dominates the function's behavior as $x \to \infty$ or $x \to -\infty$.

• As $x \to \infty$, $5x^{14}$ dominates and grows very large. So, $\lim_{x \to \infty} f(x) = \infty$

• As $x \to -\infty$, $5x^{14}$ still grows very large because $x^{14}$ is positive for both positive and negative $x$. So, $\lim_{x \to -\infty} f(x) = \infty$

Limit notation:

$\lim_{x \to \infty} f(x) = \infty, \quad \lim_{x \to -\infty} f(x) = \infty$

Would you like further clarification on any part of this, or additional details?

Related Questions:

1. How do polynomial functions behave as $x \to \pm \infty$ depending on their leading term?
2. How do lower-degree terms affect the end behavior of a polynomial function?
3. What role does the coefficient of the leading term play in determining the direction of the end behavior?
4. Can you explain how to identify the dominant term in any polynomial function?
5. How does symmetry affect the end behavior of even-degree polynomials like $x^{14}$?

Tip:

The degree and sign of the leading term in a polynomial give the most crucial information for determining the function's end behavior.