The limit expression provided is:

$\lim_{x \to \infty} \frac{8x + 3}{9x - 4}$

To find the limit as $x \to \infty$:

1. Step 1: Divide by the highest power of $x$:
   The highest power of $x$ in both the numerator and denominator is $x$. We will divide both the numerator and the denominator by $x$:

   $\frac{8x + 3}{9x - 4} = \frac{\frac{8x}{x} + \frac{3}{x}}{\frac{9x}{x} - \frac{4}{x}} = \frac{8 + \frac{3}{x}}{9 - \frac{4}{x}}$

2. Step 2: Evaluate the limit as $x \to \infty$:
   As $x$ approaches infinity, the terms $\frac{3}{x}$ and $\frac{4}{x}$ approach 0. Therefore, the expression simplifies to:

   $\lim_{x \to \infty} \frac{8 + \frac{3}{x}}{9 - \frac{4}{x}} = \frac{8 + 0}{9 - 0} = \frac{8}{9}$

Conclusion:

$\lim_{x \to \infty} \frac{8x + 3}{9x - 4} = \frac{8}{9}$

Let me know if you need further clarification or details!

Related questions:

1. What happens if the degree of the numerator is higher than the denominator in a limit problem?
2. Can you explain how limits involving infinity relate to horizontal asymptotes?
3. How do you evaluate limits when the variable approaches negative infinity?
4. What happens if both the numerator and denominator approach 0 as $x \to \infty$?
5. How would the result change if the highest power of $x$ was different in the numerator and denominator?

Tip:

When dealing with rational functions and limits at infinity, always divide each term by the highest power of $x$ to simplify the expression.