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.

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!