at x>0 |x|/x =1
therefor limit at x>0 is 1
at x<0 |x|/x=-1
therfor limit at x<0 is -1
1 and -1 are the right hand limit and left hand limit at x=0 respectively.
since they are not equal therefor at x=0 limit don't exist.
at x>0 |x|/x =1
therefor limit at x>0 is 1
at x<0 |x|/x=-1
therfor limit at x<0 is -1
1 and -1 are the right hand limit and left hand limit at x=0 respectively.
since they are not equal therefor at x=0 limit don't exist.
Edit:
limit when $\displaystyle x\rightarrow 0^{+}$
limit when $\displaystyle x\rightarrow 0^{-}$