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Question Number 141932 by mathmax by abdo last updated on 24/May/21

$$\mathrm{calculate}{\lim }_{\mathrm{x}\to 0}\frac{\sin \left(\sin \left(\mathrm{sinx}\right)\right)+1-\cos \left({\mathrm{x}}^2\right)}{{\mathrm{x}}^3}$$

Answered by TheSupreme last updated on 24/May/21

$$sin\left(sin\left(sin\left(x\right)\right)\right)+1-cos\left(x^2\right)\to x+\frac{x^4}{2}$$$$solim=\mathrm{\infty }$$

Answered by mathmax by abdo last updated on 26/May/21

$$\mathrm{sinx}\sim \mathrm{x}\Rightarrow \sin (\sin \left(\mathrm{sinx}\right)\sim \mathrm{x}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{\sin (\sin \left(\mathrm{sinx}\right)+1-{\mathrm{cosx}}^2}{{\mathrm{x}}^3}$$$$\sim \frac{\mathrm{x}+1-\cos \left({\mathrm{x}}^2\right)}{{\mathrm{x}}^3}=\frac{1}{{\mathrm{x}}^2}+\frac{1-\cos \left({\mathrm{x}}^2\right)}{{\mathrm{x}}^2}\sim \frac{1}{{\mathrm{x}}^2}+\frac{{\mathrm{x}}^4}{{2\mathrm{x}}^2}=\frac{1}{{\mathrm{x}}^2}+\frac{{\mathrm{x}}^2}{2}\Rightarrow$$$${\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)=+\mathrm{\infty }$$$$$$

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