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Question Number 80172 by Rio Michael last updated on 31/Jan/20

$$Giventhat\underset{x\to 0}{\lim }\frac{\sqrt{f\left(x\right)+x}}{h}=Lthen$$$$\underset{x\to 0}{\lim }\frac{\sqrt{f\left(x\right)+2x}}{h}=?$$

Commented by john santu last updated on 31/Jan/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{f\left(x\right)+x}}{h}or\underset{x\to 0}{\lim }\frac{\sqrt{f\left(x\right)+x}}{x}?$$

Commented by john santu last updated on 31/Jan/20

$$weassume$$$$\underset{x\to 0}{\lim }\frac{\sqrt{f\left(x\right)+x}}{x}=\underset{x\to 0}{\lim }\sqrt{\frac{f\left(x\right)+x}{x^2}}=L$$$$\sqrt{\underset{x\to 0}{\lim }\frac{f\left(x\right)+x}{x^2}}=L$$$$\underset{x\to 0}{\lim }\frac{f{}^{\prime }\left(x\right)+1}{2x}=L^2,(f{}^{\prime }\left(0\right)=-1)$$$$\underset{x\to 0}{\lim }\frac{f{}^{″}\left(x\right)}{2}=L^2\Rightarrow f{}^{″}\left(0\right)=2L^2$$$$similarly\underset{x\to 0}{\lim }\frac{\sqrt{f\left(x\right)+2x}}{x}=L$$$$$$

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