Assertion (A) : f(x) =`xsin ((1)/(x))` is differentiable at x=0 Reason (R): F(x)is continuous at x=0

Options

A. Both A and R are individually true, and R is the correct explanation of A
B. Both A and R are individually true but R is not the correct explanation of A.
C. A is true but R is false.
D. A is false but R is true.

Answer

D

Explanation

Given function : `f(x)=xsin""((1)/(x))`
For differentiablity at x=0 LHD=RHD at x=0
`LHD=underset(hto0)lim(-h)sin(-(1)/(h))/(-h)=underset(hto0)lim(hsin((1)/(h)))/(-h)`
`=underset(hto0)limsin((1)/(h))=a`
finite value lies between -1 and 1 which cannot be qualified exactly.
`RHD=underset(hto0)lim(f(0+h)-f(0))/(h)`
`=underset(hto0)lim(hsin((1)/(h)))/(h)=underset(hto0)limsin((1)/(h))`
= a finite value lies between -1 and 1 which cannot qualified exactly.
`LHDneRHDnea` definite value.
Hence, f(x) is not differentiable at x=0. For continuity at x=0 :
`underset(xto0)limLHL=underset(xto0)limRHL=V.F" at "x=0`
`LHL=underset(xto0)limf(0-h)=underset(hto0)lim-hsin(-(1)/(h))`
`=underset(hto0)limhsin""(1)/(h)=0`
`RHL=underset(xto0)lim,f(0+h)=underset(hto0)limhsin""(1)/(h)=0`
f(0)=0
`impliesLHL=RHL=f(0)`
Hence, f(x) is continuous at x=0 Thus, (A) is false but (R) is true.