Question

Discuss the continuity of the following function, at x = 0.

`f(x)=x/|x|,for x ne0`

`=1,`for `x=0`

Answer 1

f(0)=1............(given)............(1)

for` x>0, |x|=x`

`lim_(x->0^+)f(x)=lim_(x->0^+)x/|x|`

`=lim_(x->0^+)x/x`

`=lim_(x->0^+)(1)`

`=1`

for x<0,|x|=-x

`lim_(x->0^-)f(x)=lim_(x->0^+)x/|x|`

`=lim_(x->0^-)-x/x`

`=lim_(x->0^-)(-1)`

`=-1`

`therefore lim_(x->0^+)f(x)nelim_(x->0^-)f(x)`

f is discontinuous at x = 0

here `lim_(x->0^+)f(x)nelim_(x->0^-)f(X)`

`therefore lim_(x->0)f(x) `does not exist

hence, it is discontinuous at x = 0