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etudier-l-existance-de-limite-en-0-0-f-x-y-x-2-y-x-y-




Question Number 167782 by SANOGO last updated on 24/Mar/22
etudier l′existance de limite en (0,0)  f(x,y)=((x^2 y)/(x+y))
$${etudier}\:{l}'{existance}\:{de}\:{limite}\:{en}\:\left(\mathrm{0},\mathrm{0}\right) \\ $$$${f}\left({x},{y}\right)=\frac{{x}^{\mathrm{2}} {y}}{{x}+{y}} \\ $$
Answered by Mathspace last updated on 25/Mar/22
x=rcosθ and y=rsinθ  ⇒((x^2 y)/(x+y))=((r^2 cos^2 θ.rsinθ)/(r(cosθ+sinθ)))  =r^2  ((cos^2 θ)/(cosθ +sinθ))→0  (r→0) ⇒  lim_((x,y)→(0,0)) f(x,y)=0
$${x}={rcos}\theta\:{and}\:{y}={rsin}\theta \\ $$$$\Rightarrow\frac{{x}^{\mathrm{2}} {y}}{{x}+{y}}=\frac{{r}^{\mathrm{2}} {cos}^{\mathrm{2}} \theta.{rsin}\theta}{{r}\left({cos}\theta+{sin}\theta\right)} \\ $$$$={r}^{\mathrm{2}} \:\frac{{cos}^{\mathrm{2}} \theta}{{cos}\theta\:+{sin}\theta}\rightarrow\mathrm{0}\:\:\left({r}\rightarrow\mathrm{0}\right)\:\Rightarrow \\ $$$${lim}_{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {f}\left({x},{y}\right)=\mathrm{0} \\ $$
Commented by SANOGO last updated on 25/Mar/22
merci
$${merci} \\ $$

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