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Question-131443

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Question Number 131443 by rs4089 last updated on 04/Feb/21
Answered by Olaf last updated on 05/Feb/21
((xy)/(y−x^2 )) = ((rcosθ.rsinθ)/(rsinθ−r^2 cos^2 θ)) = (((1/2)rsin(2θ))/(sinθ−rcos^2 θ)) lim_(r→0) ((xy)/(y−x^2 )) = 0
$$\frac{{xy}}{{y}−{x}^{\mathrm{2}} }\:=\:\frac{{r}\mathrm{cos}\theta.{r}\mathrm{sin}\theta}{{r}\mathrm{sin}\theta−{r}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$=\:\frac{\frac{\mathrm{1}}{\mathrm{2}}{r}\mathrm{sin}\left(\mathrm{2}\theta\right)}{\mathrm{sin}\theta−{r}\mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$\underset{{r}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{xy}}{{y}−{x}^{\mathrm{2}} }\:=\:\mathrm{0} \\ $$

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