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Question Number 184798 by Spillover last updated on 11/Jan/23

Show that   lim_(x→0)  (x/(∣x∣))  does not exist

$${Show}\:{that}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}}{\mid{x}\mid}\:\:{does}\:{not}\:{exist} \\ $$

Commented by MJS_new last updated on 11/Jan/23

x→0^− : (x/(∣x∣))=(x/(−x))=−1  x→0^+ : (x/(∣x∣))=(x/x)=+1

$${x}\rightarrow\mathrm{0}^{−} :\:\frac{{x}}{\mid{x}\mid}=\frac{{x}}{−{x}}=−\mathrm{1} \\ $$$${x}\rightarrow\mathrm{0}^{+} :\:\frac{{x}}{\mid{x}\mid}=\frac{{x}}{{x}}=+\mathrm{1} \\ $$

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