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?

Question

Let $f : R \to R$ be a function. We say that $f$ has

PROPERTY 1: If $\lim_{h \to 0} \frac{f (h) - f (0)}{\sqrt{| h |}}$ exists and is finite, and

PROPERTY 2: If $\lim_{h \to 0} \frac{f (h) - f (0)}{h^{2}}$ exists and is finite.

Then which of the following options is/are correct? [2019]

Smart Achievers · Accepted Answer

(1, 3)

Property 1: $\lim_{h \to 0} \frac{f (h) - f (0)}{\sqrt{| h |}}$ exists and is finite

Property 2: $\lim_{h \to 0} \frac{f (h) - f (0)}{h^{2}}$

(1) $f (x) = x^{2 / 3}$ for Property 1

$\lim_{h \to 0} \frac{h^{2 / 3} - 0}{\sqrt{| h |}} = \lim_{h \to 0} \frac{| h |^{2 / 3}}{| h |^{1 / 2}} = \lim_{h \to 0} | h |^{1 / 6} = 0$

$∴$ $option (1) is correct.$

(2) $f (x) = \sin x$ for Property 2

$\lim_{h \to 0} \frac{\sin h - \sin 0}{h^{2}} = \lim_{h \to 0} \frac{\sin h}{h} \times \frac{1}{h}$ which does not exist.

$∴$ $(2) is incorrect option.$

(3) $f (x) = | x |$ for Property 1

$\lim_{h \to 0} \frac{| h | - 0}{\sqrt{| h |}} = \lim_{h \to 0} \sqrt{| h |} = 0$

$∴$ $option (3) is correct.$

(4) $f (x) = x | x |$ for Property 2

$\lim_{h \to 0} \frac{h | h | - 0}{h^{2}} = \lim_{h \to 0} \frac{| h |}{h}$

LHL $= - 1$ and RHL $= 1$

$∴$ $\lim_{h \to 0} \frac{| h |}{h} does not exist$

$∴$ $option (4) is incorrect.$

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