Which of the following is the correct algebraic method to find the limit $\underset{x\to 2}{lim}\frac{x^2-4}{x-2}$?

• A. Apply L’Hôpital’s Rule by differentiating numerator and denominator.
• B. Multiply numerator and denominator by $(x+2)$.
• C. Factor the numerator as $(x-2)(x+2)$, cancel $(x-2)$, then substitute $x=2$.
• D. Substitute $x=2$ directly into the expression.

Answer: (C)

Verified step by step guidance

1. Step 1: Recognize that the given limit lim_{x → 2} (x^2 - 4)/(x - 2) involves a rational function where direct substitution of x = 2 results in an indeterminate form (0/0). This indicates that simplification is required.
2. Step 2: Factor the numerator x^2 - 4 using the difference of squares formula: x^2 - 4 = (x - 2)(x + 2). Rewrite the expression as ((x - 2)(x + 2))/(x - 2).
3. Step 3: Cancel the common factor (x - 2) in the numerator and denominator, leaving the simplified expression x + 2. Note that this cancellation is valid only for x ≠ 2.
4. Step 4: Substitute x = 2 into the simplified expression x + 2 to evaluate the limit. This avoids the indeterminate form and provides the correct value of the limit.
5. Step 5: Conclude that the correct algebraic method is to factor the numerator, cancel the common term, and then substitute x = 2 into the simplified expression.