Find the limit.
$\lim_{x\rarr-2}\frac{x^2-5x-14}{x+2}$

• A. $-9$
• B. $-2$
• C. $0$
• D. Does not exist

Answer: (A)

Verified step by step guidance

1. Identify the limit expression: \( \lim_{x \to -2} \frac{x^2 - 5x - 14}{x + 2} \).
2. Notice that direct substitution of \( x = -2 \) in the denominator \( x + 2 \) results in zero, indicating a potential indeterminate form.
3. Factor the numerator \( x^2 - 5x - 14 \). Look for two numbers that multiply to \(-14\) and add to \(-5\). These numbers are \(-7\) and \(2\), so the factorization is \((x - 7)(x + 2)\).
4. Rewrite the limit expression using the factorization: \( \lim_{x \to -2} \frac{(x - 7)(x + 2)}{x + 2} \).
5. Cancel the common factor \( x + 2 \) from the numerator and the denominator, resulting in \( \lim_{x \to -2} (x - 7) \). Now, substitute \( x = -2 \) to find the limit.