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Question

If f:RR is defined by f(x)=cos3xcosxx2,forx0λ,,forx=0 and if f is continuous at x=0, then λ is equal to

A
2
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B
4
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C
6
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D
8
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Solution

The correct option is A 4
Given that,
f(x)=cos3xcosxx2forx0λforx=0
Now, LHL =limx0f(x)
=limx0cos3xcosxx2
=limh0cos3(0h)cos0h0h2
=limh0cos3hcoshh2
=limh03sin3h+sinh2h (using L'Hospitals' rule)
=9+12=4
Since, f(x) is continuous at x=0
limx0f(x)=f(0)
4=λ
λ=4

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