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प्रश्न
Discuss the continuity of the following function, at x = 0.
`f(x)=x/|x|,for x ne0`
`=1,`for `x=0`
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उत्तर
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
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