If `f(x)={(1-cosx)/(xsinx),x!=0 and 1/2,x=0` then at `x=0,f(x)` is (a)continuous and differentiable (b)differentiable but not continuous (c)continuous but not differentiable (d)neither continuous nor differentiable

Explanation

`f(x)={{: ((1-cosx)/(xsinx)),if xne0),(1/2,if x=0):}`

(L.H.L at x=0)=` lim_(x->0^-)f(x)`

`= lim_(h->0)f(0-h)`

`=lim_(h->0) (1-cos(0-h))/((0-h)sin(0-h))`

`=lim_(h->0) (1-cosh)/(hsinh)xxh/h`

`=1/2`

(R.H.L at x=0)=` lim_(x->0^+)f(x)`

`= lim_(h->0)f(0+h)`

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