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Question

Let f(x)=(1cos xx2, if x01, if x=0. Which of the following is/are true?

A
f(x) will be continuous at x = 0 if f(x) = 12 at x = 0
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B
f(x) has a removable discontinuity.
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C
limx0 f(x) exists.
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D
f(x) has a non-removable discontinuity
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Solution

The correct option is C limx0 f(x) exists.

f(x)=(1cos xx2, if x01, if x=0(LHL at x=0)=limx0 f(x)=limh0 f(0h)=limh0 1cos (h)(h)2=limh0 1cos (h)h2=limh0 2 sin2 (h2)h2=limh0 2 sin2 (h2)4(h2)2=12limh0 sin (h2)(h2)2=12(RHL at x=0)=limx0+ f(x)=limh0 f(0+h)=limh0 1cos hh2=limh0 2 sin2 (h2)h2=limh0 2 sin2 (h2)4(h2)2=12limh0 sin (h2)(h2)2=12(LHL at x=0)=(RHL at x=0)f(0)limx0 f(x) exists.f(x) has removable discontinuity at x=0.f(x) will be continuous if f(0)=12

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