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If f(x)={((x^(6)-(1)/(64))/(x^(3)-(1)/(8...

If `f(x)={((x^(6)-(1)/(64))/(x^(3)-(1)/(8))",", x ne (1)/(2)), (k"," , x =(1)/(2)):} ` is continuous at `x=(1)/(2)` then the value of k is

A

`(1)/(2)`

B

`(1)/(3)`

C

`(1)/(4)`

D

`(1)/(5)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given `f(x)={((x^(6)-(1)/(64))/(x^(3)-(1)/(8))",",x =(1)/(2)),(K",",x=(1)/(2)):}`
Since, f(x) is continuous at `x=(1)/(2)`
`therefore f((1)/(2))=lim_(x to (1)/(2))f(x)`
`rArr K=lim_(x to (1)/(2)) (x^(6)-(1)/(64))/(x^(3)-(1)/(8))=lim_(x to (1)/(2)) (6x^(5))/(3x^(2))`
`=2 lim_(x to (1)/(2)) x^(3)=2 xx((1)/(2))^(2)=(1)/(4)`
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