Determine if f defined by`f(x)={x^2sin1/x , ifx!=0 0, ifx=0`

Explanation

`f(x) = {{:(x^(2)"sin"(1)/(x)","if x ne0),(0"," if x = 0):}`
at x = 0, f(0) = 0
L.H.L. `underset(x to 0^(-))(lim)f(x)`
Let 0 -h = x
`rArr 0 - h to 0`
`rArr h to 0`
`underset(h to 0)(lim)f(0 - h)`
`underset(h to 0)(lim)(0 - h)^(2) "sin "(1)/(0 - h)`
`0 xx a finite value = 0 `
R.H.L. `underset(x to 0^(+))(lim)f(x)`
Let 0 + h =x
`rArr 0 - h to 0`
`rArr h to 0`
`underset(h to 0)(lim)f(0 + h)`
`underset(h to 0)(lim)(0 + h)^(2) "sin "(1)/(0 + h)`
`0xx` a finite value = 0
`:' L.H.L. = f(0)` = R.H.L
`:. f(x)` is continuous at x = 0