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To solve this limit problem using L'Hopital's Rule, let's break it down step by step:

### Step 1: Understanding the problem

You are given the limit:

\[
\lim_{x \to \infty} \frac{5x^2 - 7x}{9x^2 + 4}
\]

This limit, at first glance, results in an indeterminate form of \(\frac{\infty}{\infty}\), which means we can apply **L'Hopital's Rule**. L'Hopital's Rule says that if you have an indeterminate form, you can differentiate the numerator and the denominator separately, then re-evaluate the limit.

### Step 2: Apply L'Hopital's Rule

Differentiate the numerator and denominator:

- The numerator is \(5x^2 - 7x\), and its derivative is:
  \[
  \frac{d}{dx}(5x^2 - 7x) = 10x - 7
  \]

- The denominator is \(9x^2 + 4\), and its derivative is:
  \[
  \frac{d}{dx}(9x^2 + 4) = 18x
  \]

Now, applying L'Hopital's Rule:

\[
\lim_{x \to \infty} \frac{10x - 7}{18x}
\]

### Step 3: Simplify the expression

Divide both terms in the numerator by \(x\):

\[
\lim_{x \to \infty} \frac{10x - 7}{18x} = \lim_{x \to \infty} \frac{10 - \frac{7}{x}}{18}
\]

As \(x \to \infty\), the term \(\frac{7}{x} \to 0\), so the limit becomes:

\[
\frac{10}{18} = \frac{5}{9}
\]

### Step 4: Matching the correct choice

Now, we match this final result \(\frac{5}{9}\) to the given choices in the question.

The correct option based on the simplification is **D**:

\[
\lim_{x \to \infty} \left( \frac{5}{9} - \frac{7x}{4} \right)
\]

because this form simplifies to \(\frac{5}{9}\) as \(x\) approaches infinity, and the subtraction term does not affect the limit significantly since \( \lim_{x \to \infty} \frac{7x}{4} \to 0\).

### Final Answer:
The correct answer is **D**.

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Do you need further details on any step or have any questions? 

Here are 5 related questions to expand your understanding:
1. What are the conditions for applying L'Hopital's Rule?
2. How would the limit change if the highest degree of \(x\) in the numerator were lower than the denominator?
3. What happens if the terms in the numerator and denominator grow at different rates?
4. How does dividing terms by \(x\) simplify evaluating limits as \(x\) approaches infinity?
5. Can you always use L'Hopital's Rule when dealing with \(\frac{\infty}{\infty}\)?

**Tip**: Always verify if the original limit gives an indeterminate form before applying L'Hopital's Rule!

Use L'Hopital's Rule to rewrite the given limit so that it is not an indeterminate form.

Learn how to apply L'Hopital's Rule to solve limits involving rational functions. This problem simplifies a limit expression by differentiating the numerator and denominator, and evaluating the final result. Suitable for advanced high school calculus students.

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