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Evaluate-lim-x-0-x-x-




Question Number 80718 by TawaTawa last updated on 05/Feb/20
Evaluate:     lim_(x→0)   (x/(∣x∣))
$$\mathrm{Evaluate}:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{x}}{\mid\mathrm{x}\mid} \\ $$
Commented by mind is power last updated on 05/Feb/20
∣x∣= { ((x,        x>0)),((−x   ,x<0)) :}  ⇒lim_(x→0^+ ) (x/(∣x∣))=lim_(x→0^+ ) =(x/x)=1  lim_(x→0^− ) (x/(∣x∣))=lim_(x→0^− ) (x/(−x))=−1  lim_(x→0)  (x/(∣x∣))  dont exist
$$\mid{x}\mid=\begin{cases}{{x},\:\:\:\:\:\:\:\:{x}>\mathrm{0}}\\{−{x}\:\:\:,{x}<\mathrm{0}}\end{cases} \\ $$$$\Rightarrow\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{{x}}{\mid{x}\mid}=\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}=\frac{{x}}{{x}}=\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\frac{{x}}{\mid{x}\mid}=\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\frac{{x}}{−{x}}=−\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}}{\mid{x}\mid}\:\:{dont}\:{exist} \\ $$
Commented by TawaTawa last updated on 08/Feb/20
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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