Question Number 5232 by sanusihammed last updated on 02/May/16
$${If}\:{f}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} −\mathrm{4}}{{x}−\mathrm{2}}\:\:\:\: \\ $$$$\:\:\:\:{x}\rightarrow\mathrm{2} \\ $$$$ \\ $$$$\alpha\:=\:\mathrm{3}\:.\:\Sigma\:=\:\mathrm{0}.\mathrm{1}\:\:\:{find}\:\:\:\delta \\ $$$$ \\ $$$${Alpha}\:=\:\mathrm{3}\:\:\:{Epsalum}\:=\:\mathrm{0}.\mathrm{1}\:{Find}\:{delta} \\ $$
Commented by FilupSmith last updated on 02/May/16
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{definition}\:\mathrm{of}\:\delta? \\ $$
Answered by Yozzii last updated on 02/May/16
$${According}\:{to}\:{the}\:\epsilon−\delta\:{definition}\:{of} \\ $$$${the}\:{limit},\:{the}\:{limit}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}{f}\left({x}\right)=\alpha \\ $$$${exists}\:{if},{for}\:{any}\:\epsilon>\mathrm{0}\:{there}\:{is}\:{a}\:\delta>\mathrm{0}, \\ $$$${such}\:{that}\: \\ $$$$\mid{f}\left({x}\right)−\alpha\mid<\epsilon\:\:{whenever}\:\:\mathrm{0}<\mid{x}−{a}\mid<\delta. \\ $$$${So}\:{using}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} −\mathrm{4}}{{x}−\mathrm{2}}\:{and}\:\alpha=\mathrm{4}\:\:\:\:\left\{{not}\:\mathrm{3}\:{since}\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\left({x}+\mathrm{2}\right)\left({x}−\mathrm{2}\right)}{{x}−\mathrm{2}}=\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\left({x}+\mathrm{2}\right)=\mathrm{2}+\mathrm{2}=\mathrm{4}\neq\mathrm{3}.\right\} \\ $$$$\Rightarrow\mid\frac{{x}^{\mathrm{2}} −\mathrm{4}}{{x}−\mathrm{2}}−\mathrm{4}\mid=\mid\frac{{x}^{\mathrm{2}} −\mathrm{4}−\mathrm{4}{x}+\mathrm{8}}{{x}−\mathrm{2}}\mid \\ $$$$=\mid\frac{{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{4}}{{x}−\mathrm{2}}\mid=\mid\frac{\left({x}−\mathrm{2}\right)^{\mathrm{2}} }{{x}−\mathrm{2}}\mid \\ $$$$=\mid{x}−\mathrm{2}\mid \\ $$$${Since}\:\mid{f}\left({x}\right)−\mathrm{4}\mid<\epsilon\:{and}\:\mid{f}\left({x}\right)−\mathrm{4}\mid=\mid{x}−\mathrm{2}\mid \\ $$$$\Rightarrow\mid{x}−\mathrm{2}\mid<\epsilon\Rightarrow\:{let}\:\epsilon=\delta.\:{Hence},\:{if} \\ $$$$\epsilon=\mathrm{0}.\mathrm{1},\:\delta=\mathrm{0}.\mathrm{1}. \\ $$