We are tasked with finding the limit of the function:

$\lim_{{x \to \infty}} \frac{x^2 - 9}{9x + 3}$

Solution:

To evaluate the limit of this rational function as $x \to \infty$, we need to analyze the behavior of both the numerator and the denominator.

Step 1: Divide by the highest power of $x$

In this case, the highest power of $x$ in the numerator is $x^2$, and in the denominator it is $x$. To simplify the limit, we can divide every term in both the numerator and denominator by $x$:

$\lim_{{x \to \infty}} \frac{\frac{x^2}{x}}{\frac{9x}{x}} = \lim_{{x \to \infty}} \frac{\frac{x^2 - 9}{x}}{9 + \frac{3}{x}}$

This simplifies to:

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

Step 2: Take the limit as $x \to \infty$

As $x \to \infty$, the terms $\frac{9}{x}$ and $\frac{3}{x}$ approach 0 because dividing a constant by an increasingly large number yields 0. Therefore, the expression becomes:

$\lim_{{x \to \infty}} \frac{x}{9} = \frac{\infty}{9} = \infty$

Conclusion:

The limit of the function as $x \to \infty$ is $\infty$, which means the function grows without bound as $x$ becomes very large.

Would you like more details on the steps? Also, here are some related questions to explore further:

1. How would the limit change if the highest degree of $x$ in the numerator and denominator were the same?
2. What happens if we evaluate the limit as $x \to -\infty$?
3. How do you compute limits when the degrees of the numerator and denominator are equal?
4. How can L'Hopital's Rule be applied to this limit?
5. What are the general strategies for solving limits at infinity?

Tip: Always consider dividing the terms by the highest power of $x$ when solving limits at infinity to simplify the expression.