`f(x)=(7|x|+5x)/(7|x|-5x)` for `x!=0,f(0)=6at x=0`is

Options

A. removable discontinuity
B. discontinuity of first kind
C. discontinuity of second kind
D. None of the above

Answer

B

Explanation

We have, `f(x)={((7|x|+5x)/(7|x|+5x)",", x ne 0),(6",",x =0):}`
Here, `LHL=lim_(x to 0^(-)) f(x)=lim_(n to 0)f(0-h)`
`=lim_(n to 0)(7|0-h|+5(0-h))/(7|0-h|-5(0-h))`
`=lim_(n to 0) (7h-5h)/(7h+5h)=lim_(x to 0) (2h)/(12h)=(1)/(6)`
`and RHL=lim_(n to 0^(+))f(x)=lim_(x to 0) f(0+h)`
`=lim_(n to 0) (7|0+h|+5(0+h))/(7|0+h|-5(0+h))`
`=lim_(n to 0) (7h+5h)/(7h-5h)=lim_(n to 0)(12h)/(2h)=lim_(n to 0) 6=6`
Clearly, `LHL ne RHL`
`therefore f(x)` has a discontinuity of first to kind.