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!

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!