To solve the limit

$\lim_{x \to \infty} \frac{4x^3(9 + 2x^2)}{(7x^3 + 5)(4 - x^2)},$

let's break down the expression step by step.

Step 1: Simplify the numerator and denominator

The numerator is:

$4x^3(9 + 2x^2) = 4x^3 \cdot 9 + 4x^3 \cdot 2x^2 = 36x^3 + 8x^5.$

The denominator is:

$(7x^3 + 5)(4 - x^2).$

Now, distribute:

$(7x^3 + 5)(4 - x^2) = 7x^3 \cdot 4 + 7x^3 \cdot (-x^2) + 5 \cdot 4 + 5 \cdot (-x^2) = 28x^3 - 7x^5 + 20 - 5x^2.$

So the expression becomes:

$\frac{36x^3 + 8x^5}{28x^3 - 7x^5 + 20 - 5x^2}.$

Step 2: Factor out the highest power of $x$

In both the numerator and denominator, the highest power of $x$ is $x^5$, so we will factor out $x^5$ from both the numerator and denominator.

• In the numerator, factor out $x^5$:

$\frac{x^5(8 + \frac{36}{x^2})}{x^5(-7 + \frac{28}{x^2} - \frac{5}{x^3} + \frac{20}{x^5})}.$

Step 3: Analyze the limit as $x \to \infty$

As $x \to \infty$, the terms with negative powers of $x$ (like $\frac{36}{x^2}$, $\frac{28}{x^2}$, $\frac{5}{x^3}$, and $\frac{20}{x^5}$) will approach zero.

So, the limit simplifies to:

$\lim_{x \to \infty} \frac{8}{-7} = -\frac{8}{7}.$

Conclusion:

The limit is

$\boxed{-\frac{8}{7}}.$

Do you need further details or have any questions?

Here are 5 related questions:

1. What happens if the highest power of $x$ in the numerator and denominator is different?
2. How do limits behave when dealing with rational functions?
3. Can limits involving infinity result in $0$, and why?
4. How does factoring help in simplifying limits at infinity?
5. What if the limit tends to zero instead of infinity? How would the approach change?

Tip: When solving limits at infinity for rational functions, always focus on the highest degree terms, as lower-degree terms become insignificant at large values of $x$.