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To evaluate this limit using L'Hôpital's Rule, we first need to confirm the limit is in an indeterminate form. Let's analyze it:

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

As \( x \to \infty \), both the numerator and the denominator approach infinity, so we can apply L'Hôpital's Rule by differentiating the numerator and the denominator separately.

### Step 1: Differentiate the numerator and the denominator
The numerator is \( 2x^2 - 7x \):
\[
\frac{d}{dx}(2x^2 - 7x) = 4x - 7
\]

The denominator is \( 7x^2 + 4 \):
\[
\frac{d}{dx}(7x^2 + 4) = 14x
\]

### Step 2: Apply L'Hôpital's Rule
Now, rewrite the limit as:
\[
\lim_{{x \to \infty}} \frac{4x - 7}{14x}
\]

### Step 3: Simplify the expression
Divide each term in the fraction by \( x \) to make the limit clearer:
\[
\lim_{{x \to \infty}} \frac{4 - \frac{7}{x}}{14} = \frac{4 - 0}{14} = \frac{4}{14} = \frac{2}{7}
\]

### Conclusion
The result of the limit is:
\[
\frac{2}{7}
\]

Would you like more details on any step or have any questions?

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Here are some related questions you might find interesting:
1. How does L'Hôpital's Rule handle higher-order polynomials in limits?
2. What other techniques are available for evaluating limits at infinity besides L'Hôpital's Rule?
3. Can L'Hôpital's Rule be applied to limits involving exponential functions?
4. How do you determine when to stop applying L'Hôpital's Rule?
5. What is the impact of different indeterminate forms (like \(\frac{0}{0}\) or \(\infty/\infty\)) on the use of L'Hôpital's Rule?

**Tip:** When you encounter an indeterminate form in a limit problem, check if L'Hôpital's Rule can simplify it by repeatedly differentiating until a determinate form is reached.

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

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

Learn how to evaluate limits at infinity using L'Hôpital's Rule with an example limit \( \lim_{{x \to \infty}} \frac{2x^2 - 7x}{7x^2 + 4} \). This step-by-step solution addresses indeterminate forms and simplifies the expression to find a determinate result.

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