Find the limit.
$\lim_{x\rarr3}\frac{x^2+2x-3}{x-2}$

• A. $0$
• B. $3$
• C. $12$
• D. $DNE$

Answer: (C)

Verified step by step guidance

1. Step 1: Identify the type of limit problem. This is a rational function limit where direct substitution of x = 3 results in an indeterminate form (0/0).
2. Step 2: Factor the numerator. The expression x^2 + 2x - 3 can be factored into (x - 1)(x + 3).
3. Step 3: Simplify the expression. Substitute the factored form into the limit expression: \( \frac{(x - 1)(x + 3)}{x - 2} \).
4. Step 4: Cancel common factors. Notice that there are no common factors between the numerator and the denominator, so the expression cannot be simplified further.
5. Step 5: Evaluate the limit using L'Hôpital's Rule. Since direct substitution leads to an indeterminate form, apply L'Hôpital's Rule by differentiating the numerator and the denominator separately and then substituting x = 3.