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Question Number 80220 by TawaTawa last updated on 01/Feb/20

Commented by kaivan.ahmadi last updated on 01/Feb/20

$$\mathrm{\forall }ϵ>0\mathrm{\exists }\delta >0s.t\mid x-3\mid <\delta \Rightarrow \mid x^2-9\mid <ϵ$$$$\mid x^2-9\mid =\mid x-3\mid \mid x+3\mid <ϵ$$$$if\delta =1\Rightarrow \mid x-3\mid <1\Rightarrow -1<x-3<1\Rightarrow 2<x<4\Rightarrow$$$$5<x+3<7\Rightarrow \mid x+3\mid <7$$$$now$$$$\mid x-3\mid <\frac{ϵ}{7}$$$$ifϵ=0.01\Rightarrow \delta <\frac{0.01}{7}=\frac{1}{700}$$$$ifϵ=0.00001\Rightarrow \delta <\frac{1}{70000}$$

Commented by TawaTawa last updated on 01/Feb/20

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by TawaTawa last updated on 01/Feb/20

$$\mathrm{How}\mathrm{did}\mathrm{you}\mathrm{come}\mathrm{across}5\mathrm{and}7\mathrm{sir}$$

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