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Let f(x)={(x-1)sin1/(x-1)if\ x!=1 0,\ if...

Let `f(x)={(x-1)sin1/(x-1)if\ x!=1 0,\ if\ x=1` . Then which one of the following is true? `f` is differentiable at `x=0\ ` and at`\ x-1` `f` is differentiable at `x=0\ ` but not at`\ x=1` `f` is differentiable at `x=0` nor at `x=1` `f` is differentiable at `x=1\ ` but not at`\ x=0`

Text Solution

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`f'(x) = lim_(h->0) (f(x+h) - f(x))/h`
`f'(0) = lim_(h-> 0) (f(h) - f(0))/h`
`lim_(h->0) ((h-1) sin(1/(h-1)) - (-1)sin(-1))/h`
`lim_(h->0) ((h-1)sin(1/(h-1)) - sin 1)/h`
`f(x)` is not differentiable at x=0
`f'(1) = lim_(h->0) (f(1+h) - f(1))/h`
`lim_(h->0) ((1+h-1)sin(1/(1+h-1)) - 0)/h`
`lim_(h->0) (hsin(1/h))/h`
...
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