https://file.aiquickdraw.com/mathbot/file/17266132019270r5lopfv.jpg

To solve the limit problem presented in the image, we need to evaluate the following limit:

\[
\lim_{x \to \infty} \frac{5x - 3}{2x + 9}
\]

### Step-by-Step Solution:

1. **Analyze the degrees of the numerator and denominator**:
   - The numerator is \(5x - 3\), and the highest degree term is \(5x\).
   - The denominator is \(2x + 9\), and the highest degree term is \(2x\).

   Since both the numerator and denominator are of degree 1 (highest degree of \(x\) is 1 in both cases), we can proceed by dividing both the numerator and denominator by \(x\), the highest power of \(x\) in the expression.

2. **Rewrite the expression**:
   \[
   \lim_{x \to \infty} \frac{5x - 3}{2x + 9} = \lim_{x \to \infty} \frac{\frac{5x}{x} - \frac{3}{x}}{\frac{2x}{x} + \frac{9}{x}}
   \]
   Simplifying this gives:
   \[
   \lim_{x \to \infty} \frac{5 - \frac{3}{x}}{2 + \frac{9}{x}}
   \]

3. **Take the limit as \(x \to \infty\)**:
   - As \(x \to \infty\), \(\frac{3}{x} \to 0\) and \(\frac{9}{x} \to 0\).
   - Thus, the expression becomes:
   \[
   \frac{5 - 0}{2 + 0} = \frac{5}{2}
   \]

### Final Answer:
\[
\lim_{x \to \infty} \frac{5x - 3}{2x + 9} = \frac{5}{2}
\]

Let me know if you would like further clarification!

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Here are 5 related questions for further exploration:
1. How do you handle limits where the degree of the numerator is higher than the degree of the denominator?
2. What is the result of a limit when the degrees of the numerator and denominator are different?
3. How would the result change if there were higher powers of \(x\) in the numerator and denominator?
4. How can you find horizontal asymptotes using limits as \(x \to \infty\)?
5. What does it mean if the limit approaches infinity instead of a finite number?

**Tip**: Always compare the degrees of the numerator and denominator when evaluating limits of rational functions at infinity!

Find the limit, if it exists: lim (x -> ∞) (5x - 3) / (2x + 9).

This page provides a detailed step-by-step solution to the limit problem lim (x -> ∞) (5x - 3) / (2x + 9). It covers important calculus concepts, including handling limits at infinity for rational functions, and is suitable for high school or college-level calculus students.

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