If `f(x)={{:(,(x-a)^(n)cos ((1)/(x-a)), x ne a),(,0,x=a):}`
then at x=a, f(x) is

Options

A. continuous if `n gt 0` and differentiable if `n gt 1`
B. continuous if `n gt 1` and differentiable if `n gt 0`
C. continuous and differentiable if `n gt 0`
D. none of these

Answer

A

Explanation

For f(x) to be continuous at x=a, we must have `underset(x to a)lim f(x)=f(a)`
`Rightarrow underset(x to a)lim (x-a)^(n) =cos ((1)/(x-a))=0`
`Rightarrow n gt 0`
For f(x) to be differentiable
`underset(x to a)lim (f(x)-f(a))/(x-a)` must exist finitely.
`Rightarrow underset(x to a)lim ((x-a)^(n) cos((1)/(x-a))-0)/(x-a)` must exist finitely
`Rightarrow (x-a)^(n) cos ((1)/(x-a))` must exist finitely
`Rightarrow n-1 gt 0`
`Rightarrow n gt 1`.
Hence, f(x) is continuous for `n gt 0` and differentiable for `n gt 1`.