`lim_(xrarr-2) (x^2-x-6)^2/(x+2)^2=`

Options

A. `6`
B. `25`
C. `9`
D. `16`

Explanation

To solve the limit \( \lim_{x \to -2} \frac{(x^2 - x - 6)^2}{(x + 2)^2} \), we will follow these steps:

### Step 1: Factor the numerator
First, we need to factor the expression in the numerator, \( x^2 - x - 6 \).

The expression can be factored as follows:
\[ x^2 - x - 6 = (x - 3)(x + 2) \]

### Step 2: Substitute the factored form into the limit
Now we can substitute the factored form back into the limit:
\[ \lim_{x \to -2} \frac{((x - 3)(x + 2))^2}{(x + 2)^2} \]

### Step 3: Simplify the expression
We can simplify the expression:
\[ \frac{((x - 3)(x + 2))^2}{(x + 2)^2} = \frac{(x - 3)^2 (x + 2)^2}{(x + 2)^2} \]
Since \( (x + 2)^2 \) in the numerator and denominator cancels out (for \( x \neq -2 \)):
\[ = (x - 3)^2 \]

### Step 4: Evaluate the limit
Now we can evaluate the limit as \( x \) approaches -2:
\[ \lim_{x \to -2} (x - 3)^2 = (-2 - 3)^2 = (-5)^2 = 25 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to -2} \frac{(x^2 - x - 6)^2}{(x + 2)^2} = 25 \]

---