Let's analyze the long-run behavior of the function:

$f(t) = 4t^9 - 3t^8 - t^3 + 3$

As $t \to -\infty$:

• The dominant term in this function as $t \to -\infty$ is the term with the highest power of $t$, which is $4t^9$.
• Since $t^9$ grows very large in magnitude as $t$ becomes negative, the behavior of $f(t)$ will be dominated by this term.
• Specifically, since $t^9$ will be negative when $t$ is negative, $4t^9$ will also be negative and grow without bound.

Thus, as $t \to -\infty$, $f(t) \to -\infty$.

As $t \to +\infty$:

• Similarly, as $t \to +\infty$, the term $4t^9$ will again dominate the behavior of the function.
• Since $t^9$ is positive for large positive $t$, $4t^9$ will grow positively without bound.

Thus, as $t \to +\infty$, $f(t) \to +\infty$.

Conclusion:

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

If you have any questions or need further clarification, feel free to ask!

Here are 5 related questions:

1. How would the function behave if the term with the highest degree were negative instead of positive?
2. What happens to the function if we add or subtract lower-degree terms?
3. How does the function behave as $t \to 0$?
4. What are the critical points of the function $f(t)$?
5. How do we determine whether a function has an inflection point?

Tip: When analyzing the long-term behavior of a polynomial, always focus on the highest-degree term since it will dominate the behavior at extreme values of $t$.