Discuss the continuity of the following functions at the points shown against them :
`{:(f(x)=(x)/(|x|)",","for"x ne0),(=1",","for"x=0):}}at x=0.`

Explanation

Given `f(x)=(x)/(|x|), "for" x ne 0`
`:. lim_(x to 0^(-))f(x) = lim _(x to 0^(-))(x)/(|x|)= lim_(x to ^(-))(x)/(-x)=-1 " "....(1)`
` lim_(x to 0^(+))f(x) = lim _(x to 0^(+))(x)/(|x|)= lim_(x to ^(+))(x)/(x)=1 " "....(2)`
From (1) and (2), we get
`:. lim _(x to o^(-)) f(x) ne lim_(x to o^(+))f(x)`
`:. lim_(x to 0)f(x)` does not exist
`:. f(x)` is discontinuous at x=0