Let f : R→ R be defined by f(x) = \{\table\x\sin(1/x),\if \x>0;0,\if \x\leq0 . Then

Correct Answer is : f is continuous but not differentiable at 0.

f is neither continuous nor differentiable at 0

f is continuous and differentiable at 0.

f is continuous but not differentiable at 0.

f is not continuous but differentiable at 0.

NIMCET 2019 Question Paper with solutions

Question : 2 of 120

Marks: +1, -0

Let f : R→ R be defined by f(x) =

\{\table\x\sin(1/x),\if\; \x>0;0,\if \;\x≤0

. Then

Solution:

We have,
f(x) =

\{\table\x\sin(1/x),\if\; \x>0;0,\if \;\x≤0

∴(R.H.L at

x=0

)

=\lim↙{x→0^{-}}x\sin(\;\;{1}/{x})=\lim↙{h→0} h\sin(\;\;{1}/{h})=0×[-1,1]=0

(LHL at

x=0

) = 0 and

f(0)=0

Hence,f(x) continuous at

x=0

Again,RHD

=\lim↙{x→0^{+}}\;\;{x\sin(\;\;{1}/{x})-0}/{x-0}=\lim↙{x→0^{+}}

\sin(\;\;{1}/{x})=\lim↙{h→0}\sin(\;\;{1}/{h})=[-1,1]

and LHD = 0
RHD ≠ LHD
Hence,f(x) is not differentiable at

x=0

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