If `f(x)={{:(,x^(m)sin((1)/(x)),x ne 0),(,0,x=0):}` is a continous at x=0, then

Options

A. `m in (0,oo)`
B. `m in (-oo,0)`
C. `m in (1,oo)`
D. `m in (-oo,1)`

Answer

A

Explanation

If f(x) is a continous at x=0, then
`underset (x to 0^(-))f(x)=underset(x to 0^(+))lim f(x)=f(0)=0`
Now,
`underset(x to 0^(-))limf(x)=underset(h to 0)lim f(0-h)=underset(h to 0)lim (-h)^(m) sin ((-1)/(h))`
`Rightarrow underset(x to 0^(-))lim f(x)=underset(h to 0)lim (-h)^(m) sin ((1)/(h))=0" only when m"gt0 and underset(x to 0^(+))limf(x)=underset(h to 0)lim f(0+h)`
`Rightarrow underset(x to 0^(+))lim f(x)=underset(x to 0) h^(m) sin ((1)/(h))=0,"only when m"gt0`
Hence, f(x) is a continous at x=0, if `m gt 0`