Find the limit.
$\lim_{x\rarr2}\frac{x^2-7x+12}{x-3}$

• A. $-1$
• B. $-2$
• C. $0$
• D. DNE

Answer: (B)

Verified step by step guidance

1. First, identify the type of limit problem. This is a rational function where both the numerator and the denominator are polynomials.
2. Check if direct substitution of x = 2 into the function results in an indeterminate form like 0/0. Substitute x = 2 into the numerator and denominator: \( x^2 - 7x + 12 \) and \( x - 3 \).
3. Calculate the numerator: \( 2^2 - 7(2) + 12 = 4 - 14 + 12 = 2 \). Calculate the denominator: \( 2 - 3 = -1 \). Since the denominator is not zero, direct substitution is possible.
4. Since direct substitution does not result in an indeterminate form, evaluate the limit by substituting x = 2 directly into the function: \( \frac{2^2 - 7(2) + 12}{2 - 3} \).
5. Simplify the expression to find the limit. The limit is the value of the function as x approaches 2.