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Question Number 118681 by bemath last updated on 19/Oct/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{x}-\sqrt{\sin x}}{x^2\sqrt{x}}=?$$

Answered by benjo_mathlover last updated on 19/Oct/20

$$\underset{x\to 0}{\lim }\frac{x-\sin x}{x^2\sqrt{x}}\times \frac{1}{\sqrt{x}+\sqrt{\sin x}}=$$$$\underset{x\to 0}{\lim }\frac{x-\sin x}{x^3}\times \frac{\sqrt{x}}{\sqrt{x}+\sqrt{\sin x}}=\frac{1}{6}\times \frac{1}{2}=\frac{1}{12}$$

Commented by Lordose last updated on 19/Oct/20

$$\mathrm{Very}\mathrm{correct}$$

Commented by 1549442205PVT last updated on 19/Oct/20

$$\underset{\mathrm{x}\to 0}{\mathrm{I}=\lim }\frac{\sqrt{x}-\sqrt{\sin x}}{x^2\sqrt{x}}.\mathrm{This}\mathrm{is}\mathrm{the}\mathrm{form}\frac{0}{0}$$$$\underset{\mathrm{x}\to 0}{=\lim }\frac{\mathrm{x}-\mathrm{sinx}}{{\mathrm{x}}^3}\times \frac{\sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{sinx}}}$$$$\underset{\mathrm{x}\to 0}{=\lim }\frac{\mathrm{x}-\mathrm{sinx}}{{\mathrm{x}}^3}\times \frac{1}{1+\sqrt{\frac{\mathrm{sinx}}{\mathrm{x}}}}=\mathrm{A}\times \frac{1}{2}$$$$\mathrm{A}{=}^{\frac{0}{0}}\underset{\mathrm{x}\to 0}{\lim }\frac{1-\mathrm{cosx}}{{3\mathrm{x}}^2}=\underset{{\mathrm{L}}^{\prime }\mathrm{Hopital}}{{}^{\frac{0}{0}}}\mathrm{li}\underset{\mathrm{x}\to 0}{\mathrm{m}}\frac{\mathrm{sinx}}{6\mathrm{x}}$$$$=1/6\Rightarrow \mathrm{I}=\frac{1}{6}\times \frac{1}{2}=\frac{1}{12}$$

Answered by mathmax by abdo last updated on 19/Oct/20

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\sqrt{\mathrm{x}}-\sqrt{\mathrm{sinx}}}{{\mathrm{x}}^2\sqrt{\mathrm{x}}}\mathrm{changement}\sqrt{\mathrm{x}}=\mathrm{t}\mathrm{give}$$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left({\mathrm{t}}^2\right)=\frac{\mathrm{t}-\sqrt{\sin \left({\mathrm{t}}^2\right)}}{{\mathrm{t}}^5}(\mathrm{x}\to 0\Rightarrow \mathrm{t}\to 0)$$$$\sin \left({\mathrm{t}}^2\right)\sim {\mathrm{t}}^2-\frac{{\mathrm{t}}^6}{6}\Rightarrow \sqrt{\sin \left({\mathrm{t}}^2\right)}\sim \mathrm{t}\sqrt{1-\frac{{\mathrm{t}}^4}{6}}\sim \mathrm{t}(1-\frac{{\mathrm{t}}^4}{12})\Rightarrow$$$$\mathrm{f}\left({\mathrm{t}}^2\right)\sim \frac{\mathrm{t}-\mathrm{t}+\frac{{\mathrm{t}}^5}{12}}{{\mathrm{t}}^5}\Rightarrow \mathrm{f}\left({\mathrm{t}}^2\right)\sim \frac{1}{12}\Rightarrow {\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{12}$$

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