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Question

# Let f:R→R be a function defined by f(x)=⎧⎨⎩sin(x2)xif x≠0 0if x=0 Then, at x=0, f is

A
not continuous
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B
continuous but not differentiable
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C
differentiable and the derivative is not continuous
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D
differentiable and the derivative is continuous
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Solution

## The correct option is D differentiable and the derivative is continuousFor continuity, limx→0f(x)=limx→0sin(x2)x=limx→0sin(x2)x2×x=0 So the function is continuous at x=0 Checking the differentiability at x=0 Right hand derivative, R.H.D =limh→0f(0+h)−f(0)h=limh→0sin(h2)h−0h=limh→0sin(h2)h2=1 Left hand derivative, L.H.D =limh→0f(0−h)−f(0)−h=limh→0sin((−h)2)−h−0−h=limh→0sin(h2)h2=1 So, L.H.D=R.H.D then function is differentiable at x=0 and f′(0)=1 When x≠0 f′(x)=d(sin(x2)x)dx⇒f′(x)=2x2cos(x2)−sin(x2)x2⇒f′(x)=2cos(x2)−sin(x2)x2f′(x)=⎧⎪⎨⎪⎩2cos(x2)−sin(x2)x2if x≠01if x=0 Checking the continuity of f′(x), limx→0f′(x)=limx→02cos(x2)−sin(x2)x2=2−1=1 So the derivative is also continuous.

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