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.