Beregn lim(x,y)→(0,0)x2yx2+y2\lim_{(x,y)\to(0,0)}\dfrac{x^2y}{x^2+y^2}lim(x,y)→(0,0)x2+y2x2y med polarkoordinater.
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x=rcosθ, y=rsinθx=r\cos\theta,\ y=r\sin\thetax=rcosθ, y=rsinθ: r2cos2θ⋅rsinθr2=rcos2θsinθ\dfrac{r^2\cos^2\theta\cdot r\sin\theta}{r^2}=r\cos^2\theta\sin\thetar2r2cos2θ⋅rsinθ=rcos2θsinθ. Siden ∣cos2θsinθ∣≤1|\cos^2\theta\sin\theta|\le 1∣cos2θsinθ∣≤1, er ∣f∣≤r→0|f|\le r\to 0∣f∣≤r→0. Grensen er 000.
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