To describe the long-run behavior of the function $f(t) = t^9 + 5t^8 - 4t^5 - 2$, we need to analyze its limits as $t$ approaches both $\infty$ (positive infinity) and $-\infty$ (negative infinity).

As $t \to \infty$:

The highest degree term in the function is $t^9$, so as $t \to \infty$, the behavior of $f(t)$ will be dominated by this term, and all lower-degree terms become insignificant in comparison.

Thus, $f(t) \approx t^9$ as $t \to \infty$. Since $t^9$ grows without bound for large positive $t$, we have:

$\lim_{t \to \infty} f(t) = \infty$

As $t \to -\infty$:

Again, the dominant term is $t^9$, but this time, we need to consider the fact that $t^9$ will be negative for large negative values of $t$ (since 9 is an odd power). So, as $t \to -\infty$, $t^9$ goes to $-\infty$ much faster than the lower-degree terms.

Thus, $f(t) \approx t^9$ as $t \to -\infty$, and we conclude:

$\lim_{t \to -\infty} f(t) = -\infty$

Conclusion:

• As $t \to \infty$, $f(t) \to \infty$.
• As $t \to -\infty$, $f(t) \to -\infty$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do lower-degree terms affect the behavior of polynomials at finite $t$?
2. What is the difference in long-run behavior between even and odd degree polynomials?
3. How does the behavior of the leading coefficient impact the function at infinity?
4. What happens to the function when both leading and non-leading terms have negative coefficients?
5. Can we predict the inflection points of this function using calculus?

Tip: When determining the end behavior of a polynomial, focus on the highest-degree term, as it dictates the overall trend for large values of the variable.