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The limit you're tasked with solving is:

\[
\lim_{{x \to -6}} \frac{{x^2 - 36}}{{x + 6}}
\]

### Step 1: Factor the numerator
The numerator \(x^2 - 36\) is a difference of squares, and it can be factored as:

\[
x^2 - 36 = (x - 6)(x + 6)
\]

### Step 2: Simplify the expression
Substitute the factored form into the expression:

\[
\frac{{(x - 6)(x + 6)}}{{x + 6}}
\]

For \(x \neq -6\), we can cancel the common factor of \(x + 6\), giving:

\[
\frac{{(x - 6)(x + 6)}}{{x + 6}} = x - 6
\]

### Step 3: Evaluate the limit
Now, take the limit as \(x \to -6\):

\[
\lim_{{x \to -6}} (x - 6) = -6 - 6 = -12
\]

### Final Answer:
\[
\boxed{-12}
\]

Would you like further details or clarification on any of the steps?

Here are some related questions you might explore:
1. What happens if the limit approaches a different value, say \(x \to 6\)?
2. How do we handle limits involving indeterminate forms like \(0/0\)?
3. What is the general rule for factoring differences of squares?
4. What other techniques can be used to simplify limits?
5. How can limits be evaluated if factoring doesn't work?

**Tip**: Always check for factorizable patterns like difference of squares in limits involving polynomials.

Find the limit: lim(x -> -6) (x^2 - 36) / (x + 6)

Learn how to solve the limit lim(x -> -6) (x^2 - 36) / (x + 6) by factoring the difference of squares. This step-by-step solution walks through simplifying and evaluating the limit. Suitable for advanced algebra students in Grades 11-12.

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