lim┬(𝑥→−2) (𝑥+2)/(𝑥²+𝑥−6 )

To evaluate the limit:

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

### Step 1: Factor the denominator

First, let's factor the quadratic expression in the denominator:

\[
x^2 + x - 6 = (x + 3)(x - 2)
\]

So the expression becomes:

\[
\frac{x + 2}{(x + 3)(x - 2)}
\]

### Step 2: Direct substitution

Now, substitute \( x = -2 \) directly into the expression:

\[
\frac{-2 + 2}{(-2 + 3)(-2 - 2)} = \frac{0}{(1)(-4)} = 0
\]

Since the numerator becomes 0 and the denominator is not 0, the limit is:

\[
\boxed{0}
\]

### Conclusion
The limit as \( x \to -2 \) of \( \frac{x+2}{x^2 + x - 6} \) is 0.

Do you want more details on this calculation or have any other questions?

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Here are 5 related questions:
1. What if the denominator was non-factorable? How would we approach the limit?
2. How do you evaluate limits with indeterminate forms like \( \frac{0}{0} \)?
3. What is the general process for finding limits at points of discontinuity?
4. How would the limit change if the numerator was a higher degree polynomial?
5. Can this method apply to limits approaching infinity?

**Tip:** Always try factoring the denominator first when dealing with rational expressions in limits. It often simplifies the calculation!

lim┬(𝑥→−2) (𝑥+2)/(𝑥²+𝑥−6)

Learn how to evaluate the limit as x approaches -2 of the rational function (x + 2) / (x² + x - 6) by factoring the quadratic expression. This step-by-step guide simplifies complex algebraic concepts and is ideal for high school students in Grades 10-12.

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