Calculate the limit algebraically. (If an answer does not exist, enter DNE.)

lim *x*→−2

*x*2 + 8*x* + 12

*x*2 + 2*x*

To solve the limit

\[
\lim_{x \to -2} \frac{x^2 + 8x + 12}{x^2 + 2x},
\]

we will first attempt to simplify the expression by factoring both the numerator and the denominator, if possible.

### Step 1: Factor the numerator
The numerator is \( x^2 + 8x + 12 \). We factor this quadratic:

\[
x^2 + 8x + 12 = (x + 6)(x + 2).
\]

### Step 2: Factor the denominator
The denominator is \( x^2 + 2x \). We factor out the common factor \( x \):

\[
x^2 + 2x = x(x + 2).
\]

### Step 3: Simplify the expression
Now, the original expression becomes:

\[
\frac{(x + 6)(x + 2)}{x(x + 2)}.
\]

We can cancel the common factor \( (x + 2) \) from both the numerator and denominator (as long as \( x \neq -2 \)):

\[
\frac{x + 6}{x}.
\]

### Step 4: Evaluate the limit
Now, we evaluate the limit as \( x \to -2 \):

\[
\lim_{x \to -2} \frac{x + 6}{x} = \frac{-2 + 6}{-2} = \frac{4}{-2} = -2.
\]

Thus, the limit is:

\[
\boxed{-2}.
\]

Would you like more details or have any further questions?

Here are 5 related questions:
1. How do you handle limits that result in indeterminate forms like \( \frac{0}{0} \)?
2. What is the definition of a limit in calculus?
3. How do you factor quadratics quickly?
4. What are common techniques to evaluate limits without factoring?
5. How do you evaluate limits that approach infinity?

**Tip:** Always check if an expression can be simplified by factoring or canceling terms before directly substituting values in limits.

Calculate the limit algebraically. lim x→−2 (x^2 + 8x + 12) / (x^2 + 2x)

Learn how to solve the limit lim x→−2 (x^2 + 8x + 12) / (x^2 + 2x) algebraically by factoring and simplifying rational expressions. This step-by-step guide covers limits, factoring quadratics, and the application of limit laws. Suitable for Grades 10-12.

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