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Question:

Let f: R → R be a function. We say that f has PROPERTY 1 if limh→0 [f(h) - f(0)]/√(|h|) exists and is finite, PROPERTY 2 if limh→0 [f(h) - f(0)]/h2 exists and is finite. Then which of the following option(s) is/are correct?

f(x) = |x| has property 1

f(x) = sin x has property 2

f(x) = x|x| has property 2

f(x) = x2/3 has property 1

Solution:

Correct option is C.

f(x) = |x| has property 1
(1) limh→0 (h2/3 - 0)/√(|h|) = 0
(2) limx→0 (sin h - 0)/h2 does not exist
(3) limh→0 (|h| - 0)/√(|h|) = 0
f(x) = x|x| ⇒ limh→0 [f(h) - f(0)]/h2 = limh→0 (h|h| - 0)/h2 does not exist