Question Bank • QB_EM_ADV_027
MCQ • 1 Mark
Section 1: Engineering Mathematics
Calculus
Subtopic: Limit, continuity and differentiability
Question Statement:
Consider the function $f(x) = |x| \sin(x)$ defined for all $x \in \mathbb{R}$. At $x = 0$, the function is:
Options:
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(A) Continuous and differentiable with $f'(0) = 0$
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(B) Continuous but not differentiable
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(C) Neither continuous nor differentiable
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(D) Continuous and differentiable with $f'(0) = 1$
Step-by-Step Solution & Explanation
Evaluate the derivative at $x = 0$ from first principles:
$f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{|h| \sin(h)}{h}$
- Right-hand limit ($h \to 0^+$): $\lim_{h \to 0^+} \frac{h \sin(h)}{h} = \lim_{h \to 0^+} \sin(h) = 0$
- Left-hand limit ($h \to 0^-$): $\lim_{h \to 0^-} \frac{-h \sin(h)}{h} = \lim_{h \to 0^-} -\sin(h) = 0$
Since the left and right limits are equal to $0$, $f'(0) = 0$. Hence $f(x)$ is continuous and differentiable at $x = 0$.
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