Question Bank • QB_EM_CALC_002
MCQ • 1 Mark
Section 1: Engineering Mathematics
Calculus
Subtopic: Limit, continuity and differentiability
Question Statement:
For the function $f(x) = |x|$ defined on $\mathbb{R}$, at the point $x = 0$, the function is:
Options:
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(A) Continuous and differentiable
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(B) Continuous but not differentiable
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(C) Differentiable but not continuous
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(D) Neither continuous nor differentiable
Step-by-Step Solution & Explanation
1. Continuity at $x = 0$:
$\lim_{x \to 0^+} |x| = 0, \quad \lim_{x \to 0^-} |x| = 0, \quad f(0) = 0$
Since the left and right limits both equal $f(0)$, $f(x)$ is continuous at $x = 0$.
2. Differentiability at $x = 0$:
$\text{Right-hand derivative: } \lim_{h \to 0^+} \frac{|0+h| - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1$
$\text{Left-hand derivative: } \lim_{h \to 0^-} \frac{|0+h| - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$
Since the right-hand derivative ($+1$) does not equal the left-hand derivative ($-1$), $f(x)$ is not differentiable at $x = 0$.
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