Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).
limit, start subscript, x, right arrow, infinity, end subscript, start fraction, 4, x, cubed, left parenthesis, 9, plus, 2, x, squared, right parenthesis, divided by, left parenthesis, 7, x, cubed, plus, 5, right parenthesis, left parenthesis, 4, minus, x, squared, right parenthesis, end fraction
x→∞
lim
​	
  
(7x 
3
 +5)(4−x 
2
 )
4x 
3
 (9+2x 
2
 )
​	
 

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 \(x\).

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).
lim_{x \to \infty} \frac{4x^3(9 + 2x^2)}{(7x^3 + 5)(4 - x^2)}

Learn how to calculate the limit of a rational function as x approaches infinity. This detailed solution covers the simplification process, factoring out the highest powers of x, and applying limit theorems to find the result of -8/7. Suitable for high school and early college students.

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