To solve the limit problem given in the image:

$\lim_{x \to 3} \frac{x^2 - x - 12}{x + 3}$

we can follow these steps:

Step 1: Simplify the expression

First, factor the numerator $x^2 - x - 12$.

The numerator factors as: $x^2 - x - 12 = (x - 4)(x + 3)$

Now, substitute this into the original limit:

$\frac{(x - 4)(x + 3)}{x + 3}$

Step 2: Cancel the common factors

Since $x + 3$ appears in both the numerator and the denominator, we can cancel it:

$\frac{(x - 4)\cancel{(x + 3)}}{\cancel{x + 3}} = x - 4, \quad \text{for } x \neq -3$

Step 3: Evaluate the limit

Now, substitute $x = 3$:

$\lim_{x \to 3} (x - 4) = 3 - 4 = -1$

Final Answer:

$\lim_{x \to 3} \frac{x^2 - x - 12}{x + 3} = -1$

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What happens if you try to evaluate the limit directly without simplifying?
2. How do you handle limits when factoring doesn’t simplify the expression?
3. What is L'Hôpital's rule, and when is it useful for finding limits?
4. How would the solution change if the limit approached $-3$ instead of $3$?
5. Can you find the limit $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$?

Tip: Always check for common factors in the numerator and the denominator when simplifying rational expressions for limits.