Question

Evaluate the limit: \[ \lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{1}{x^2 - 3x + 2} \right]. \]

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

When simplifying limits with common terms in the numerator and denominator, factor the expression and cancel out common factors.

Answer 1

The Correct Option is D

We are given: \[ \lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{1}{x^2 - 3x + 2} \right]. \] Factor the denominator of the second fraction: \[ x^2 - 3x + 2 = (x - 1)(x - 2). \] Thus, the expression becomes: \[ \lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{1}{(x - 1)(x - 2)} \right]. \] Now, find a common denominator: \[ = \lim_{x \to 2} \frac{(x - 1) - 1}{(x - 2)(x - 1)} = \lim_{x \to 2} \frac{x - 2}{(x - 2)(x - 1)}. \] Canceling \( (x - 2) \) from the numerator and denominator, we get: \[ = \lim_{x \to 2} \frac{1}{x - 1}. \] Substitute \( x = 2 \): \[ = \frac{1}{2 - 1} = 1. \]