Question

$\footnotesize \rm Let \: f :\R\to\R \: be \: a \: function. \: We \: say \: that \: f \: has \\ \footnotesize\rm PROPERTY \: 1 \: if \: \lim_{h \to0} \frac{f(h) - f(0)}{ \sqrt{ |h| } } \: exixts \: and \: is \: finite, and \\ \footnotesize \rm PROPERTY \: 2 \: if\lim_{h \to0} \frac{f(h) - f(0)}{ {h}^{2} } \: exists \: and \: if \: finite. \\ \footnotesize \rm Then \: which \: of \: the \: following \: options \: is \: correct? \\ \footnotesize \rm A. \: f(x) = x|x| \: has \: PROPERTY \: 2 \\ \footnotesize \rm B. \: f(x) = sinx \: has \: PROPERTY \: 2 \\ \footnotesize \rm C. \: f(x) = |x| \: has \: PROPERTY \: 1 \\ \footnotesize \rm D. \: f(x) = {x}^{ \frac{2}{3} } \: has \: PROPERTY \: 1$


​

Answer 1

$\sf \fbox \red{Question}$

$\footnotesize \rm Let \: f :\R\to\R \: be \: a \: function. \: We \: say \: that \: f \: has \\ \footnotesize\rm PROPERTY \: 1 \: if \: \lim_{h \to0} \frac{f(h) - f(0)}{ \sqrt{ |h| } } \: exixts \: and \: is \: finite, and \\ \footnotesize \rm PROPERTY \: 2 \: if\lim_{h \to0} \frac{f(h) - f(0)}{ {h}^{2} } \: exists \: and \: if \: finite. \\ \footnotesize \rm Then \: which \: of \: the \: following \: options \: is \: correct? \\ \footnotesize \rm A. \: f(x) = x|x| \: has \: PROPERTY \: 2 \\ \footnotesize \rm B. \: f(x) = sinx \: has \: PROPERTY \: 2 \\ \footnotesize \rm C. \: f(x) = |x| \: has \: PROPERTY \: 1 \\ \footnotesize \rm D. \: f(x) = {x}^{ \frac{2}{3} } \: has \: PROPERTY \:$

$\: \underline{ \rule{190pt}{2pt}}$

$\sf \fbox \red{solution}$

$\sf \: correct \: option \: \\ \\ \sf (D). \: f(x) = x ^{ \frac{2}{3} } has \: property \: 1 \\ \\ \sf (C).f(x) = |x| \: has \: property \: 1$

$\: \underline{ \rule{190pt}{2pt}}$

$\sf \fbox \red{explanation }$

$\sf \: (A). \: f(x) = x|x| \\ \\ \sf \: = \lim_{h \to0} \frac{h (h) - 0}{ {h}^{2} } = \lim_{h \to0} \frac{ |h| }{h} = \binom{1 \: if \: h \to0 ^{ + } }{ - 1 \: if \: h \to0 ^{ - } } \\ \\ \sf \: so \: \lim_{h \to0} \: \frac{f(h) - f(0)}{h} dose \: not \: exist$

$\:$

______________________________________

$\sf \: (B).f(x) = \sin \: x \\ \\ \sf \: = \lim_{h \to0} \frac{ \sin(h - 0) }{ {h}^{2} } = \lim_{h \to0} \frac{1}{h} . \frac{ \sin(h) }{h} \: dose \: not \: exist$

$\:$

______________________________________

$\sf \: (C).f(x) = |x| \\ \\ \sf \: = \lim_{h \to0} \frac{ |h| - 0}{ \sqrt{ |h| } } = \lim_{h \to0} \sqrt{ |h| } = 0$

$\:$

______________________________________

$\sf \:( D).f(x) = {x}^{ \frac{2}{3} } \\ \\ \sf \: = \lim_{h \to0} \frac{ h\frac{2}{3} - 0 }{ \sqrt{ |h| } } = \lim_{h \to0} |h|^{ \frac{1}{6} } = 0$

$\: \underline{ \rule{190pt}{2pt}}$