To evaluate the limit \(\lim_{{x \to -3}} \frac{x^2 - x - 12}{x + 3}\), let's proceed with the following steps:

1. **Substitute \(x = -3\) directly** into the expression to see if we can find the limit without further simplification.

\[
\frac{(-3)^2 - (-3) - 12}{-3 + 3} = \frac{9 + 3 - 12}{0} = \frac{0}{0}
\]

Since we get an indeterminate form \(\frac{0}{0}\), we need to simplify the expression.

2. **Factor the numerator**:

The numerator \(x^2 - x - 12\) can be factored as:

\[
x^2 - x - 12 = (x - 4)(x + 3)
\]

3. **Simplify the expression**:

Now, the limit becomes:

\[
\lim_{{x \to -3}} \frac{(x - 4)(x + 3)}{x + 3}
\]

We can cancel out the \((x + 3)\) term in the numerator and the denominator (as long as \(x \neq -3\)):

\[
\lim_{{x \to -3}} (x - 4)
\]

4. **Evaluate the limit**:

Now, substitute \(x = -3\) into the simplified expression:

\[
-3 - 4 = -7
\]

Thus, the limit is:

\[
\boxed{-7}
\]

Would you like any further details or have any questions?

Here are some related questions:

1. How do you handle limits that result in other indeterminate forms like \(\frac{\infty}{\infty}\)?
2. What is L'Hôpital's Rule, and when is it used?
3. How do you find the limit of a trigonometric function as \(x\) approaches a specific value?
4. What are the methods for factoring quadratic expressions like the one in this problem?
5. How do you evaluate limits at infinity?

**Tip:** When you encounter an indeterminate form, simplifying the expression by factoring or rationalizing is often the key to finding the limit.

lim(x^2 - x - 12) / (x + 3) as x approaches -3

Discover how to evaluate the limit of (x^2 - x - 12) / (x + 3) as x approaches -3. This step-by-step guide explains factoring quadratics and handling indeterminate forms like 0/0. Ideal for Grades 10-12 students studying limits and algebra.

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