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Question Number 90944 by john santu last updated on 27/Apr/20

$$\underset{x\to 0}{\lim }\frac{x\sin x}{2\sin {}^{2}3x-x^2\cos x}?$$

Commented by john santu last updated on 27/Apr/20

Commented by mathmax by abdo last updated on 27/Apr/20

$$letf\left(x\right)=\frac{xsinx}{2{sin}^2\left(3x\right)-x^{2}cosx}wehavesinx\sim x\Rightarrow xsinx\sim x^2$$$$sin\left(3x\right)\sim 3x\Rightarrow 2{sin}^2\left(3x\right)\sim 18x^{2}andcosx\sim 1\Rightarrow$$$$f\left(x\right)\sim \frac{x^2}{18x^2-x^2}=\frac{1}{17}\Rightarrow {lim}_{x\to 0}f\left(x\right)=\frac{1}{17}$$

Commented by john santu last updated on 27/Apr/20

$$\mathrm{good}\mathrm{sir}$$

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