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Question

Examine the origin for continuity and derivability in case of the function f defined by f(x)=xtan1(1/x), x0 and f(0)=0.

A
continuous but not differentiable at x=0
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B
continuous and differentiable at x=0
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C
not continuous and not differentiable at x=0
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D
none of these
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Solution

The correct option is A continuous but not differentiable at x=0
f(0+)=limh0(htan1(1h))=0×π2=0
f(0)=limh0(htan1(1h))=0×π2=0
Hence the function is continuous at x=0.
f(0+)=limh0htan1(1h)h=π2
f(0+)=limh0(htan1(1h))=π2
Hence the function is not differentiable at x=0.

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