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Question Number 93287 by john santu last updated on 12/May/20

Commented by mathmax by abdo last updated on 12/May/20

$$letf\left(x\right)=\frac{arctan\left(x^2(1-cosx)\right)}{(1-\sqrt{cosx})ln\left(\frac{sinx}{x}\right)}wehave1-cosx\sim \frac{x^2}{2}\Rightarrow x^2(1-cosx)\sim \frac{x^4}{2}\Rightarrow$$$$arctan\left(x^2(1-cosx)\right)\sim arctan\left(\frac{x^4}{2}\right)\sim \frac{x^4}{2}$$$$sinx\sim x-\frac{x^3}{6}\Rightarrow \frac{sinx}{x}\sim 1-\frac{x^2}{6}\Rightarrow ln\left(\frac{sinx}{x}\right)\sim ln(1-\frac{x^2}{6})\sim -\frac{x^2}{6}$$$$cosx\sim 1-\frac{x^2}{2}\Rightarrow \sqrt{cosx}\sim \sqrt{1-\frac{x^2}{2}}\sim 1-\frac{x^2}{4}\Rightarrow 1-\sqrt{cosx}\sim \frac{x^2}{4}\Rightarrow$$$$f\left(x\right)\sim \frac{\frac{x^4}{2}}{\frac{x^2}{4}\times (-\frac{x^2}{6})}=-\frac{4\times 6}{2}=-12\Rightarrow {lim}_{x\to 0}f\left(x\right)=-12$$$$$$

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