Let `f(x)={:{(x^nsin.(1)/x,","xne0),(0,","x=0):}`, then f(x) is continuous but not differentiable at x = 0, if

Options

A. `n in (0,1)`
B. `n in [1,oo)`
C. `(-oo,0)`
D. `n=0`

Answer

A

Explanation

Since , f(x) is continuous at x = 0 , therefore
`=underset(xto0)limf(x)= f(0) rArr underset(xto0)limx^(n) sin""(1)/(x)= 0`, which is
possible when ` n gt 0 `
f(x) is differentiable ` at x = 0 if underset(x to 0)lim (f(x) - f(0))/(x-0)` exists finitely .
`rArr underset(x to 0) lim (f(x) -f(0))/(x-0) ` exists finitely .
`rArr underset(x to 0) lim x^(n-1) ""sin(1)/(x)` exists finitely .
` rArr n - 1 gt 0 rArr n gt 1`
Hence , f(x) is continuous but not differerentiable at x = 0 , if
` n in (0,1)` .