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Let \mathrm{f(x)=[x], g(x)=|x|}  and \mathrm{ g \{f(x)\}=h(x)}, where [ \cdot ]  is the greatest integer function. Then \mathrm{h^{\prime}(-1)}  is
 

Option: 1

0


Option: 2

-\infty


Option: 3

 nonexistent


Option: 4

 none of these


Answers (1)

best_answer

\begin{aligned} & \mathrm{h(x)=f\{g(x)\}=[g(x)]=[|x|] .} \\ & \begin{aligned} \mathrm{h^{\prime}(-1+0)} & \mathrm{=\lim _{h \rightarrow 0} \frac{[|-1+h|]-[|-1|]}{h}=\lim _{h \rightarrow 0} \frac{[1-h]-[1]}{h}=\lim _{h \rightarrow 0} \frac{0-1}{h}=-\infty . }\\ \mathrm{h^{\prime}(-1-0) }& =\mathrm{\lim _{h \rightarrow 0} \frac{[|-1-h|]-[|-1|]}{-h}=\lim _{h \rightarrow 0} \frac{[1+h]-[1]}{-h} }\\ & \mathrm{=\lim _{h \rightarrow 0} \frac{1-1}{-h}=\lim _{h \rightarrow 0}=0 .} \end{aligned} \end{aligned}

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Rakesh

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