To determine the limits of the function $f(x) = \frac{5x + 1}{x^2 + 7x - 18}$ as $x$ approaches certain values, let's analyze the behavior of the function around those points.

First, let's factor the denominator to identify potential issues or points where the function might be undefined:

$x^2 + 7x - 18 = (x + 9)(x - 2)$

This shows that the function has vertical asymptotes at $x = -9$ and $x = 2$. These are the points where the denominator is zero and the function may go to $\pm \infty$.

Given Answers:

1. $\lim_{x \to 3^-} f(x) = \frac{4}{3}$
2. $\lim_{x \to 3^+} f(x) = \frac{4}{3}$

Both these answers are correct.

Checking Remaining Limits:

3. $\lim_{x \to 2^-} f(x)$
4. $\lim_{x \to 2^+} f(x)$

Since $x = 2$ is a vertical asymptote, let's determine the behavior of the function as $x$ approaches 2 from the left and from the right.

Limit as $x \to 2^-$:

As $x$ approaches 2 from the left ($x \to 2^-$): $f(x) = \frac{5x + 1}{(x + 9)(x - 2)}$

When $x$ is slightly less than 2, the term $(x - 2)$ is negative and very small, causing the fraction to become very large in magnitude but negative in sign: $\lim_{x \to 2^-} \frac{5x + 1}{(x + 9)(x - 2)} = -\infty$

So, the correct answer is $-\infty$.

Limit as $x \to 2^+$:

As $x$ approaches 2 from the right ($x \to 2^+$): $f(x) = \frac{5x + 1}{(x + 9)(x - 2)}$

When $x$ is slightly greater than 2, the term $(x - 2)$ is positive and very small, causing the fraction to become very large in magnitude but positive in sign: $\lim_{x \to 2^+} \frac{5x + 1}{(x + 9)(x - 2)} = \infty$

So, the correct answer is $\infty$.

Summary of Correct Answers:

1. $\lim_{x \to 3^-} f(x) = \frac{4}{3}$
2. $\lim_{x \to 3^+} f(x) = \frac{4}{3}$
3. $\lim_{x \to 2^-} f(x) = -\infty$
4. $\lim_{x \to 2^+} f(x) = \infty$

Would you like more details on any of these steps or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you determine the horizontal asymptotes of a rational function?
2. What is the significance of the numerator in the limit calculation for rational functions?
3. How do you find the oblique asymptotes of a function?
4. How can you use limits to find the points of discontinuity in a function?
5. What is the difference between removable and non-removable discontinuities?

Tip: When evaluating limits approaching a vertical asymptote, always check the signs of the terms involved to determine if the limit goes to $\infty$ or $-\infty$.