Question Detail

MTH 102 Differentiation Objective Question
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Question

If \( \lim_{x \to 2} \frac{f(x)}{x-2} = f'(2) = 0 \), which must be true?
Options
A I only: \( f(2) = 0 \)
B II only: \( f(x) \) is continuous at \( x=2 \)
C III only: \( f(x) \) has a horizontal tangent at \( x=2 \)
D I, II, and III are all true
Correct Answer
Correct Answer

Option D is the correct answer.

Detailed Explanation

I: The limit exists, so ( f(2) ) must be 0 for the form 0/0. II: Differentiability implies continuity. III: ( f'(2)=0 ) means slope is 0.

Hint

Recall the definitions of differentiability, continuity, and the slope of a tangent line.

Question Info

Course: MTH 102
Topic: Differentiation
Type: Objective
Difficulty: Not specified
Question 111 of 930
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