1.

Let f: R be a function. We say that f has PROPERTY 2 if lim_(hto0)(f(h)-f(0))/(h^2) exists and is finite. Then which of the following options is/are correct ?

Answer»

`f(x)=sinx` has PROPERTY 2
`f(x)=x^(2//3)` has PROPERTY 1
`f(x)=|x|` has PROPERTY 1
`f(x)x|x|` has PROPERTY 2

Solution :it is given, that `f:rtoR` and property 1, `underset(hto0)(lim)(f(h)-f(0))/(SQRT(|h|))` exists and finite, and
property 2, `underset(hto0)(lim)(f(h)-(f(0)))/(h^(2))` exists and finite.
OPTION a,
`P2:underset(hto0)(lim)(h^(2//3)-0)/(sqrt(|h|))=underset(hto0)(lim)(1)/(h)((sinh)/(h))=`doesn't exist.
option b,
`P1:underset(hto0)(lim)(|h|-0)/(sqrt(|h|))=underset(hto0)(lim)(|h|)/(h)={{:(1,"if "hto0^(+)),(-1," if "hto0^(-)):}`
so `underset(hto0)(lim)(f(h)-f(0))/(h^(2))` does not exist.


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