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To find the limit:

\[
\lim_{{x \to \infty}} \frac{5x^2 - 3x}{7x^2 + 1}
\]

we can analyze the behavior of the function as \( x \to \infty \). Here’s the step-by-step solution:

### Step 1: Identify the highest power of \( x \) in the numerator and denominator
In this case, the highest power in both the numerator and the denominator is \( x^2 \).

### Step 2: Divide both the numerator and the denominator by \( x^2 \)
This gives us:

\[
\lim_{{x \to \infty}} \frac{\frac{5x^2 - 3x}{x^2}}{\frac{7x^2 + 1}{x^2}} = \lim_{{x \to \infty}} \frac{5 - \frac{3}{x}}{7 + \frac{1}{x^2}}
\]

### Step 3: Take the limit as \( x \to \infty \)
As \( x \to \infty \), the terms \( \frac{3}{x} \) and \( \frac{1}{x^2} \) approach zero. So, we are left with:

\[
\lim_{{x \to \infty}} \frac{5 - 0}{7 + 0} = \frac{5}{7}
\]

### Final Answer
\[
\lim_{{x \to \infty}} \frac{5x^2 - 3x}{7x^2 + 1} = \frac{5}{7}
\]

Would you like further details on any part of the solution, or do you have any other questions?

Here are some related questions to explore:
1. How would the limit change if the power of \( x \) in the numerator was higher than in the denominator?
2. What is the limit if the powers of \( x \) are the same but the coefficients change?
3. How does L'Hôpital's Rule apply to problems with limits approaching infinity?
4. What if we approach \( x \to -\infty \) instead of \( x \to \infty \)?
5. How can you use polynomial division to solve similar limits?

**Tip:** For rational functions, when \( x \to \infty \), comparing the highest power terms in the numerator and denominator helps determine the behavior of the limit.

Learn how to find the limit of (5x^2 - 3x) / (7x^2 + 1) as x approaches infinity. This solution involves dividing by the highest power of x and understanding the behavior of limits at infinity in rational functions. Suitable for advanced high school students.

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