Question Number 158341 by aliibrahim1 last updated on 02/Nov/21
Answered by puissant last updated on 03/Nov/21
$${That}\:{is}\:\varepsilon>\mathrm{0},\:{let}'{s}\:{seek}\:\alpha>\mathrm{0}\:/\:\forall{x}\in\mathbb{R}\:,\:\mathrm{0}<\mid{x}−\mathrm{2}\mid<\alpha\:\Rightarrow\:\mid{x}^{\mathrm{2}} −\mathrm{4}\mid<\varepsilon \\ $$$${for}\:{x}\in\mathbb{R}\:,\:\mid{x}^{\mathrm{2}} −\mathrm{4}\mid\:=\:\mid{x}−\mathrm{2}\mid.\mid{x}+\mathrm{2}\mid\:{we}\:{have}\:\mid{x}−\mathrm{2}\mid<\mathrm{2}\:\rightarrow\:\mathrm{0}<{x}+\mathrm{2}<\mathrm{6} \\ $$$$\Rightarrow\:\mid{x}^{\mathrm{2}} −\mathrm{4}\mid<\:\mathrm{6}\mid{x}−\mathrm{2}\mid\:;\:\mid{x}^{\mathrm{2}} −\mathrm{4}\mid<\varepsilon\:\rightarrow\:\mathrm{6}\mid{x}−\mathrm{2}\mid<\varepsilon\:\rightarrow\:\mid{x}−\mathrm{2}\mid<\frac{\varepsilon}{\mathrm{6}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{take}\:\alpha={min}\left(\mathrm{2}\:;\:\frac{\varepsilon}{\mathrm{6}}\right)....\blacksquare \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.................\mathscr{L}{e}\:{puissant}............... \\ $$
Commented by aliibrahim1 last updated on 04/Nov/21
$${thx}\:{sir} \\ $$