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Question Number 182508 by peter frank last updated on 10/Dec/22

Answered by mahdipoor last updated on 10/Dec/22

$$limf\left(x\right),x\to c=F$$$$\iff \mathrm{\forall }{ϵ}_1>0\mathrm{\exists }{\delta }_1>0\mid x-c\mid <{\delta }_1\Rightarrow \mid F-f\left(x\right)\mid <{ϵ}_1$$$$limg\left(x\right),x\to c=G$$$$\iff \mathrm{\forall }{ϵ}_2>0\mathrm{\exists }{\delta }_2>0\mid x-c\mid <{\delta }_2\Rightarrow \mid G-g\left(x\right)\mid <{ϵ}_2$$$$,$$$$\mathrm{\forall }\epsilon ={ϵ}_1+{ϵ}_2\mathrm{\exists }\delta =max({\delta }_1,{\delta }_2)$$$$\mid x-c\mid <\delta \Rightarrow$$$$\mid (F+G)-(f\left(x\right)+g\left(x\right))\mid ⩽\mid F-f\left(x\right)\mid +\mid G-g\left(x\right)\mid <{ϵ}_1+{ϵ}_2=\epsilon$$$$\iff$$$$lim(f+g),x\to c=F+G$$$$$$

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