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The given limit is:

\[
\lim_{x \to \infty} \frac{8x^2 - 3}{7x^2 + x - 3}
\]

### Step-by-step solution:

1. **Factor the highest degree of \(x\)** from both the numerator and the denominator. The highest power of \(x\) in this case is \(x^2\).

   \[
   \frac{8x^2 - 3}{7x^2 + x - 3} = \frac{x^2(8 - \frac{3}{x^2})}{x^2(7 + \frac{1}{x} - \frac{3}{x^2})}
   \]

2. **Simplify** the expression by canceling out \(x^2\) from both the numerator and denominator:

   \[
   = \frac{8 - \frac{3}{x^2}}{7 + \frac{1}{x} - \frac{3}{x^2}}
   \]

3. **Evaluate the limit** as \(x \to \infty\). Notice that as \(x\) becomes larger, terms like \(\frac{1}{x}\) and \(\frac{1}{x^2}\) approach 0.

   - \(\frac{3}{x^2} \to 0\)
   - \(\frac{1}{x} \to 0\)

   Thus, the expression simplifies to:

   \[
   \frac{8 - 0}{7 + 0 - 0} = \frac{8}{7}
   \]

### Final Answer:

\[
\lim_{x \to \infty} \frac{8x^2 - 3}{7x^2 + x - 3} = \frac{8}{7}
\]

Would you like more details or have any questions? Here are some related questions to explore:

1. What happens if the degree of the numerator is lower than the denominator when finding limits at infinity?
2. How does dividing each term by the highest power of \(x\) help in simplifying limits at infinity?
3. What are the limits of functions as \(x \to -\infty\)?
4. Can we apply the same method for rational functions with higher degrees, like \(x^3\)?
5. How would the limit change if we had a linear term in both the numerator and denominator?

**Tip:** When evaluating limits at infinity, always check the highest power of \(x\) in both the numerator and the denominator—this helps to simplify the expression quickly!

lim (x -> ∞) [(8x^2 - 3) / (7x^2 + x - 3)]

Solve the limit of the rational function (8x^2 - 3) / (7x^2 + x - 3) as x approaches infinity. Learn the step-by-step process, including simplifying rational functions and canceling the highest powers of x. Perfect for students in Grades 11-12 studying calculus or pre-calculus.

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