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Question Number 115318 by bemath last updated on 25/Sep/20

$$\underset{x\to 0}{\lim }\frac{x\sin x}{2\sin {}^2\left(3x\right)-x^2\cos x}$$

Answered by bobhans last updated on 25/Sep/20

$$\underset{x\to 0}{\lim }\frac{x\sin x}{2\sin {}^2\left(3x\right)-x^2\cos x}=$$$$\underset{x\to 0}{\lim }\frac{x^2\left(\frac{\sin x}{x}\right)}{2x^2{\left(\frac{\sin 3x}{x}\right)}^2-x^2\cos x}=$$$$\underset{x\to 0}{\lim }\frac{x^2}{x^2(2.9-1)}=\frac{1}{17}$$

Answered by Bird last updated on 25/Sep/20

$$letf\left(x\right)=\frac{xsinx}{2{sin}^2\left(3x\right)-x^{2}cosx}$$$$wehavexsinx\sim x^2$$$${sin}^2\left(3x\right)\sim 9x^2$$$$x^{2}cosx\sim x^2(1-\frac{x^2}{2})\Rightarrow$$$$f\left(x\right)\sim \frac{x^2}{18x^2-x^2+\frac{x^4}{2}}=\frac{x^2}{17x^2+\frac{x^4}{2}}$$$$f\left(x\right)\sim \frac{1}{17+\frac{x^2}{2}}\Rightarrow {lim}_{x\to 0}f\left(x\right)=\frac{1}{17}$$

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