Let F : R → R be a function. We say that f has: PROPERTY 1 if lim h → 0 ⁡ f h - f 0 h exists and is finite, and PROPERTY 2 if lim h → 0 ⁡ f h - f 0 h 2 exists and is finite. Then which of the following options is/are correct?

Step-by-step solution

A f x = x lim h → 0 ⁡ h - 0 h = lim h → 0 ⁡ h = 0 B f x = x 2 3 lim h → 0 ⁡ h 2 3 - 0 h = lim h → 0 ⁡ h 1 6 = 0 C f x = x x lim h → 0 ⁡ h h - 0 h 2 = lim h → 0 ⁡ h h = 1 , i f h → 0 + - 1 , i f h → 0 - Does not exist D f x = sin ⁡ x lim h → 0 ⁡ sin ⁡ h - 0 h 2 = lim h → 0 ⁡ 1 h sin ⁡ h h ⇒ does not exist