lim x kuadrat - 4x - 12 per x kuadrat + 3x + 2 = lim x adalah -2

Let's analyze the limit:
\[
\lim_{x \to -2} \frac{x^2 - 4x - 12}{x^2 + 3x + 2}.
\]

### Step 1: Substitute \(x = -2\)
To check if we can directly substitute \(x = -2\):
\[
\frac{(-2)^2 - 4(-2) - 12}{(-2)^2 + 3(-2) + 2} = \frac{4 + 8 - 12}{4 - 6 + 2} = \frac{0}{0}.
\]
This results in an indeterminate form \( \frac{0}{0} \), so we need to simplify the expression.

### Step 2: Factor the numerator and denominator
Factor both the numerator and the denominator:
- Numerator: \(x^2 - 4x - 12 = (x - 6)(x + 2)\).
- Denominator: \(x^2 + 3x + 2 = (x + 2)(x + 1)\).

The expression now looks like this:
\[
\lim_{x \to -2} \frac{(x - 6)(x + 2)}{(x + 2)(x + 1)}.
\]

### Step 3: Cancel the common factor
We can cancel the \((x + 2)\) term (since it's not zero when \(x \neq -2\)):
\[
\lim_{x \to -2} \frac{x - 6}{x + 1}.
\]

### Step 4: Substitute \(x = -2\) again
Now, substitute \(x = -2\) into the simplified expression:
\[
\frac{-2 - 6}{-2 + 1} = \frac{-8}{-1} = 8.
\]

### Conclusion
The limit is:
\[
\lim_{x \to -2} \frac{x^2 - 4x - 12}{x^2 + 3x + 2} = 8.
\]

Would you like more details or have any questions about this solution?

### Related Questions
1. What happens when the limit results in an indeterminate form like \( \frac{0}{0} \)?
2. How do you factor quadratic expressions like \(x^2 - 4x - 12\)?
3. What is the meaning of canceling terms in a rational expression?
4. How do limits work when approaching infinity?
5. What other methods can be used to solve indeterminate forms besides factoring?

### Tip
Always check for indeterminate forms first when dealing with limits to decide the best method for simplification.

lim x^2 - 4x - 12 / x^2 + 3x + 2 = lim x → -2

Learn how to solve the limit of the quadratic expression (x^2 - 4x - 12) / (x^2 + 3x + 2) as x approaches -2. This solution involves factoring and simplifying rational expressions and is suitable for high school students in Grades 11-12.

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