Question

If \( f(x) = \frac{|x - 2|}{x - 2} \) for \( x \neq 2 \), and \( f(x) = 1 \) for \( x = 2 \), then which of the following statements is true?

• A. \( f(x) \) is continuous at \( x = 2 \)
• B. \( \lim_{x \to 2^-} f(x) = f(2) \)
• C. \( \lim_{x \to 2^+} f(x) = \lim_{x \to 2^-} f(x) \)
• D. \( f(x) \) is discontinuous at \( x = 2 \)

Hint:

To check for continuity at a point, the left-hand and right-hand limits must be equal and match the function value at that point.

Answer 1

The Correct Option is D

Step 1: Check the left-hand and right-hand limits.
For \( x \neq 2 \), the function \( f(x) = \frac{|x - 2|}{x - 2} \) is either 1 or -1 depending on whether \( x>2 \) or \( x<2 \).
Step 2: Conclusion.
Since the left-hand and right-hand limits of \( f(x) \) are not equal at \( x = 2 \), the function is discontinuous at \( x = 2 \). Therefore, the correct answer is (D) \( f(x) \) is discontinuous at \( x = 2 \).