Let `fRtoR` be a function we say that f has
property 1 if `underset(hto0)(lim)(f(h)-f(0))/(sqrt(|h|))` exist and is finite.
Property 2 if `underset(h to 0)(lim)(f(h)-f(0))/(h^(2))` exist and is finite. Then which of the following options is/are correct?

Options

A. `f(x)=x|x|` has property 2
B. `f(x)=x^(2//3)` has property 1
C. `f(x)=sinx` has property 2
D. `f(x)=|x|` has property 1

Answer

B, D

Explanation

(A) `f(X)=x|x|`
`impliesunderset(hto0)(lim)(f(h)-f(0))/(h^(2))=underset(hto0)(lim)(h|h|-0)/(h^(2))` does not exist
(B) `underset(hto0)(lim)(h^(2//3)-0)/(sqrt(|h|))=0`
(C) `underset(hto0)(lim)(sin h-0)/(h^(2))` does not exist
(D) `underset(hto 0)(lim)(|h|-0)/(sqrt(|h|))=0`