Let f:ℝ→ℝ be a function. We say that f has PROPERTY 1 if limh→0fh−f0hexists and is finite, and PROPERTY 2 if limh→0fh−f0h2 exists and is finite Then which of the following options is/are correct ?

If f(x)=|x|⇒limh→0|h|-0|h|=limh→0|h|=0 If f(x)=sinx⇒limh→0sinh-0h2=D·E If f(x)=x|x|=limh→0h|h|-0h2=D·E If f(x)=x2/3⇒limh→0h2/3-0|h|1/3=limn→0|h|1/3=0

Let f:ℝ→ℝ be a function. We say that f has PROPERTY 1 if limh→0fh−f0hexists and is finite, and PROPERTY 2 if limh→0fh−f0h2 exists and is finite Then which of the following options is/are correct ?

If f(x)=|x|⇒limh→0|h|-0|h|=limh→0|h|=0 If f(x)=sinx⇒limh→0sinh-0h2=D·E If f(x)=x|x|=limh→0h|h|-0h2=D·E If f(x)=x2/3⇒limh→0h2/3-0|h|1/3=limn→0|h|1/3=0

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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 ?

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$f (x) = |x|$ has property 1

$f (x) = \sin x$ has property 2

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

$f (x) = x^{2 / 3}$ has property 1

answer is A, D.

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Detailed Solution

$If f (x) = | x | \Rightarrow \lim_{h \to 0} \frac{| h | - 0}{\sqrt{| h |}} = \lim_{h \to 0} \sqrt{| h |} = 0$

$If f (x) = \sin x \Rightarrow \lim_{h \to 0} \frac{\sinh - 0}{h^{2}} = D \cdot E$

$If f (x) = x | x | = \lim_{h \to 0} \frac{h | h | - 0}{h^{2}} = D \cdot E$

$If f (x) = x^{2 / 3} \Rightarrow \lim_{h \to 0} \frac{h^{2 / 3} - 0}{| h |^{1 / 3}} = \lim_{n \to 0} | h |^{1 / 3} = 0$

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Let f:ℝ→ℝ be a function. We say that f has PROPERTY 1 if limh→0fh−f0hexists and is finite, and PROPERTY 2 if limh→0fh−f0h2 exists and is finite Then which of the following options is/are correct ?

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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 ?