Examine the origin for continuity and derivability in case of the function f defined by f(x)=xtan−1(1/x), x≠0 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+)=limh→0(htan−1(1h))=0×π2=0 f(0−)=limh→0(−htan−1(1−h))=0×−π2=0 Hence the function is continuous at x=0. f′(0+)=limh→0htan−1(1h)h=π2 f(0+)=limh→0(−htan−1(1−h))=−π2 Hence the function is not differentiable at x=0.