If `f(x)={(x^(k) sin((1)/(x))",",x ne 0),(0",", x =0):}` is continuous at x = 0, then

Options

A. contiuous
B. differentiable
C. continuous but not differentiable
D. None of the above

Answer

C

Explanation

Given `, f ( x ) = {{:(x sin ((1 ) / (x ) ),",", x ne 0 ) , ( 0 , ",", x = 0 ) :} `
For continuity at x = 0
LHL = ` f ( 0 - 0 ) = lim _ ( hto 0 ) f ( 0 - h ) `
` = lim _ ( hto 0 ) ( - h ) sin ( - ( 1 ) /( h) ) `
` = 0 xx ` ( finite quantity )
= 0
RHL = ` f ( 0 + 0 ) =lim_ ( hto 0 ) f ( 0 + h ) `
` = lim _ ( hto 0 ) ( h ) sin ((1 )/ ( h)) `
= 0 `xx ` ( finite quantity )
= 0
and ` f ( 0 ) = 0 `
` therefore f (0 ) = LHL = RHL `
` therefore f (x ) ` is continuous at x = 0
For differentiability at ` x = 0 `
` Rf' (0 ) = lim _ ( hto 0 ) ( f ( 0 + h ) - f ( 0 ) ) / ( h ) `
` = lim _ ( hto 0) ( h sin (( 1 )/( h) ) - 0 )/( h ) `
` = lim _ ( h to 0 ) sin (( 1 )/ ( h ) ) `
= ( a finite quantity persist between -1 to + 1 )
= does not exist
` because Rf ' ( 0 ) ne L f ' ( 0 ) `
` therefore f ( x ) ` is not differentiable at ` x = 0 `