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Question Number 180438 by Mastermind last updated on 12/Nov/22

$$\mathrm{li}\underset{\mathrm{x}\to 0}{\mathrm{m}}\frac{\sqrt{4+\mathrm{x}}-\sqrt{4-\mathrm{x}}}{\mathrm{x}}$$$$$$$$\mathrm{find}\mathrm{the}\mathrm{limit}\mathrm{above}$$

Answered by Frix last updated on 12/Nov/22

$${\mathrm{l}}^{\prime }\mathrm{H}\widehat{\mathrm{o}pital}$$$$=\underset{x\to 0}{\lim }\frac{\frac{1}{2\sqrt{4+x}}+\frac{1}{2\sqrt{4-x}}}{1}=\frac{1}{2}$$

Commented by Mastermind last updated on 12/Nov/22

$$\mathrm{I}\mathrm{also}\mathrm{got}1/2$$$$\mathrm{thanks}$$$$$$

Answered by mr W last updated on 12/Nov/22

$$=2\underset{x\to 0}{\lim }\frac{\sqrt{1+\frac{x}{4}}-\sqrt{1-\frac{x}{4}}}{x}$$$$=2\underset{x\to 0}{\lim }\frac{(1+\frac{1}{2}\times \frac{x}{4}+o\left(x^2\right))-(1-\frac{1}{2}\times \frac{x}{4}+o\left(x^2\right))}{x}$$$$=2\underset{x\to 0}{\lim }(\frac{1}{4}+o\left(x\right))$$$$=2\times \frac{1}{4}=\frac{1}{2}$$$$$$$$or$$$$=\underset{x\to 0}{\lim }\frac{(\sqrt{4+x}-\sqrt{4-x})(\sqrt{4+x}+\sqrt{4-x})}{x(\sqrt{4+x}+\sqrt{4-x})}$$$$=\underset{x\to 0}{\lim }\frac{2}{\sqrt{4+x}+\sqrt{4-x}}$$$$=\frac{2}{\sqrt{4}+\sqrt{4}}=\frac{1}{2}$$

Commented by Mastermind last updated on 12/Nov/22

$$\mathrm{Wow}!\mathrm{thank}\mathrm{you}\mathrm{man}$$$$$$

Answered by cortano1 last updated on 12/Nov/22

$$\mathrm{L}=\underset{x\to 0}{\lim }\frac{2\mathrm{x}}{\mathrm{x}(\sqrt{4+\mathrm{x}}+\sqrt{4-\mathrm{x}})}$$$$=\frac{2}{\sqrt{4}+\sqrt{4}}=\frac{2}{4}=0.5$$

Commented by Mastermind last updated on 12/Nov/22

$$\mathrm{Thank}\mathrm{you}\mathrm{boss}$$

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