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Question Number 114710 by mohammad17 last updated on 20/Sep/20

Answered by 1549442205PVT last updated on 20/Sep/20

$$\mathrm{We}\mathrm{need}\mathrm{find}\delta >0\mathrm{such}\mathrm{that}\mathrm{\forall }\mathrm{x},\mathrm{y}:$$$$\mid \mathrm{x}-0\mid <\delta ,\mid \mathrm{y}-0\mid <\delta \left(1\right)\mathrm{then}$$$$\mid {\mathrm{x}}^2+{\mathrm{y}}^2-0\mid <8\left(2\right).\mathrm{Because}$$$$\mathrm{It}\mathrm{is}\mathrm{clear}\mathrm{when}\mid \mathrm{x}\mid <1\mathrm{and}\mid \mathrm{y}\mid <1$$$$\mathrm{then}\mid {\mathrm{x}}^2+{\mathrm{y}}^2\mid =2<8\mathrm{hence},\mathrm{choosing}\delta =1$$$$\mathrm{then}\mathrm{for}\mathrm{\forall }\mathrm{x},\mathrm{y}\mathrm{satisfy}\left(1\right)\mathrm{the}\mathrm{inequality}$$$$\left(2\right)\mathrm{is}\mathrm{alway}\mathrm{true}.\mathrm{Therefore},\mathrm{in}\mathrm{the}$$$$\mathrm{available}\mathrm{answers}\mathrm{we}\mathrm{choose}\mathrm{a}$$$$\left(\mathrm{the}\mathrm{other}\mathrm{options}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{satisfy}\right)$$

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