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in Continuity and Differentiability by (91.5k points)
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Show that `f(x)=|x|` is not differentiable at `x=0.`

1 Answer

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Best answer
`f(x)=|x|`
`lim_(h->0) (f(x+h)-f(x))/h`
`LHL=RHL`
`LHL`
when `x<0 , x=0`<br> `f(x)=-x`
`lim_(h->0) (f(0+h)-f(0))/h `
`=(-h-0)/h`
`=-1`
`RHL`
when`x>0`
`lim_(h->0) (f(0+h)-f(0))/h`
`=(h+0)/h=1`
`LHL=-1,RHL=1`
`LHL&RHL`Both exits but not equal
`LHL!=RHL`

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