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Question Number 70025 by Tony Lin last updated on 30/Sep/19

$$use\epsilon -\delta defintiontoprovethat$$$$\underset{x\to 0}{\lim }\frac{\sqrt{1+x}-\sqrt{1-x}}{x}=1$$

Commented by mind is power last updated on 01/Oct/19

$$=li\underset{t\to 0}{m}\frac{\sqrt{1+sin\left(t\right)}-\sqrt{1-sin\left(t\right)}}{sin\left(t\right)}$$$$=\underset{x\to 0}{{\lim }_{}}\frac{1}{cos\left(\frac{t}{2}\right)}$$$$\mathrm{\forall }\epsilon >0\mathrm{\exists }\eta >0$$$$\mid x\mid ⩽\eta \Rightarrow \mid f\left(x\right)-1\mid ⩽\epsilon$$$$1-\epsilon ⩽\frac{1}{cos\left(\frac{t}{2}\right)}⩽1+\epsilon$$$$\Rightarrow \frac{1}{1-\epsilon }⩾cos\left(\frac{t}{2}\right)⩾\frac{1}{1+\epsilon }$$$$\Rightarrow \frac{t}{2}⩽arcos(\frac{1}{1-\epsilon )}$$$$\Rightarrow t⩽{2\cos }^{-1}\left(\frac{1}{1-\epsilon }\right)={\eta }_{\epsilon }$$$$$$$$$$$$$$$$$$$$$$$$$$

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