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Question Number 112497 by bemath last updated on 08/Sep/20

$$\mathrm{If}\underset{x\to 0}{\lim }\frac{\sin 2\mathrm{x}+\mathrm{asin}\mathrm{x}}{{\mathrm{x}}^3}\mathrm{exist}$$$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{a}\mathrm{and}\mathrm{the}\mathrm{limit}?$$

Answered by mathmax by abdo last updated on 08/Sep/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)=\frac{\sin \left(2\mathrm{x}\right)+\mathrm{asinx}}{{\mathrm{x}}^3}\mathrm{we}\mathrm{have}\mathrm{sinx}\sim \mathrm{x}-\frac{{\mathrm{x}}^3}{6}$$$$\sin \left(2\mathrm{x}\right)\sim 2\mathrm{x}-\frac{{\left(2\mathrm{x}\right)}^3}{6}=2\mathrm{x}-\frac{4}{3}{\mathrm{x}}^3\Rightarrow \mathrm{f}\left(\mathrm{x}\right)\sim \frac{2\mathrm{x}-\frac{4}{3}{\mathrm{x}}^3+\mathrm{ax}-\frac{{\mathrm{ax}}^3}{6}}{{\mathrm{x}}^3}$$$$=\frac{2+\mathrm{a}-{\mathrm{x}}^2(\frac{4}{3}+\frac{\mathrm{a}}{6})}{{\mathrm{x}}^2}\mathrm{so}{\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)\mathrm{exist}\iff 2+\mathrm{a}=0\iff \mathrm{a}=-2$$$$\mathrm{and}\mathrm{the}\mathrm{limit}\mathrm{is}\mathrm{L}=-(\frac{4}{3}-\frac{1}{3})=-1$$

Answered by bemath last updated on 08/Sep/20

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