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Question

If f(x)=1cos xx sin x, x012, x=0 then at x = 0, f(x) is

A
continuous
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B
not continuous
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C
differentiable
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D
not differentiable
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Solution

The correct options are
A continuous
C differentiable
We have,
f(x)=1cos xx sin x, x012, x=0f(x)=⎜ ⎜2 sin2 (x/2)x (2 sin x/2 cos x/2), x012, x=0f(x)=tan x/2x , x012, x=0(LHL at x=0)=limx0f(x)=limh0f(0h)=limh0tan h/2h =12limh0 tan h/2h/2 =12(RHL at x=0)=limx0+f(x)=limh0f(0+h)=limh0tan h/2h =12limh0 tan h/2h/2 =12(LHL at x=0)=(RHL at x=0)=f(0)f(x) is continuous at x=0.(LHD at x=0)=limx0f(x)f(0)x0=limh0f(0h)f(0)0h0=limh0tan (h2)h12h =limh0tan (h2)+h2h2 =limh0⎜ ⎜ ⎜h2+(h2)33+2(h2)515....⎟ ⎟ ⎟+h2h2 =limh0(h2)33+2(h2)515....h2 =limh0h83+2h33215....= 0(RHD at x=0)=limx0+f(x)f(0)x0=limh0f(0+h)f(0)0+h0=limh0tan (h2)h12h =limh0tan (h2)h2h2 = limh0⎜ ⎜ ⎜h2+(h2)33+2(h2)515....⎟ ⎟ ⎟h2h2= limh0(h2)33+2(h2)515....h2= limh0h83+2h33215....= 0(LHD at x=0)=(RHD at x=0)f(x) is differentiable at x=0

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