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Question Number 122106 by bemath last updated on 14/Nov/20

$$\underset{x\to 0}{\lim }\frac{3\sin \left(\pi x\right)-\sin \left(3\pi x\right)}{x^3}=?$$

Answered by liberty last updated on 14/Nov/20

$$\underset{x\to 0}{\lim }\frac{3(\pi \mathrm{x}-\frac{{\pi }^3{\mathrm{x}}^3}{6})-(3\pi \mathrm{x}-\frac{27{\pi }^3{\mathrm{x}}^3}{6})}{{\mathrm{x}}^3}$$$$\underset{x\to 0}{\lim }\frac{-\frac{{\pi }^3{\mathrm{x}}^3}{2}+\frac{9{\pi }^3{\mathrm{x}}^3}{2}}{{\mathrm{x}}^3}=4{\pi }^3.▴$$

Answered by Bird last updated on 14/Nov/20

$$f\left(x\right)=\frac{3sin\left(\pi x\right)-sin\left(3\pi x\right)}{x^3}$$$$\Rightarrow f\left(x\right){=}_{\pi x=t}{\pi }^3\times \frac{3sin\left(t\right)-sin\left(3t\right)}{t^3}$$$$wehavesint\sim t-\frac{t^3}{6}$$$$sin\left(3t\right)\sim 3t-\frac{27t^3}{6}=3t-\frac{9}{2}{t}^3$$$$\Rightarrow f\left(\frac{t}{\pi }\right)\sim {\pi }^3.\frac{3t-\frac{t^3}{2}-3t+\frac{9}{2}{t}^3}{t^3}$$$$\Rightarrow f\left(\frac{t}{\pi }\right)\sim 4{\pi }^3\Rightarrow {lim}_{x\to 0}f\left(x\right)=4{\pi }^3$$

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