https://file.aiquickdraw.com/mathbot/file/17270156038708k43a8nj.jpg

The problem in the image involves finding the limit of a rational function as \( x \to \infty \). The expression is:

\[
\lim_{x \to \infty} \left( \frac{-3x^2 + 5x + 4}{4 - 2x - x^2} \cdot \frac{\frac{5x^3}{4 - x^2}}{1} \right)
\]

We can simplify this by breaking it into parts:

### Step 1: Simplifying the first term
The first term is:

\[
\frac{-3x^2 + 5x + 4}{4 - 2x - x^2}
\]

As \( x \to \infty \), the highest degree term dominates. So, we approximate:

\[
\frac{-3x^2}{-x^2} = 3
\]

### Step 2: Simplifying the second term
The second part is:

\[
\frac{5x^3}{4 - x^2}
\]

Again, as \( x \to \infty \), the highest degree term dominates:

\[
\frac{5x^3}{-x^2} = -5x
\]

### Step 3: Multiply the results
Multiplying the two results:

\[
3 \times (-5x) = -15x
\]

### Conclusion:
Since the expression grows without bound as \( x \to \infty \), the limit of the function is \( -\infty \).

Let me know if you would like further clarification!

Here are 5 related questions you might find useful:

1. What is the behavior of rational functions as \( x \to -\infty \)?
2. How do you determine which terms dominate in polynomials?
3. What is the method for finding limits involving infinity?
4. How do horizontal asymptotes relate to limits at infinity?
5. Can L'Hôpital's Rule be applied to this problem?

**Tip:** When dealing with limits at infinity, always focus on the highest-degree terms, as they dominate the behavior of the function.

Find the limit of the expression as x approaches infinity: ( (-3x^2 + 5x + 4) / (4 - 2x - x^2) ) * (5x^3 / (4 - x^2))

Learn how to find the limit of complex rational functions involving polynomials as x approaches infinity. This problem involves simplifying polynomials and analyzing their highest-degree terms. Suitable for high school calculus students.

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