Granica funkcji dwóch zmiennych

[jjjjjj]

Oblicz granicę \( \Lim_{(x,y)\to (0,0)} \frac{1- \cos (x^2+y^2)}{(x^2+y^2)x^2 y^2} \)

[Jerry]

\( \Lim_{(x,y)\to (0,0)} \frac{1- \cos (x^2+y^2)}{(x^2+y^2)x^2 y^2}= \Lim_{(x,y)\to (0,0)} \frac{1^2- \cos^2 (x^2+y^2)}{(x^2+y^2)x^2 y^2(1+ \cos (x^2+y^2))}=\\ \qquad =\Lim_{(x,y)\to (0,0)}\left(\frac{\sin (x^2+y^2)}{x^2+y^2}\right)^2\cdot\left({1\over y^2}+{1\over x^2}\right)\cdot\frac{1}{1+ \cos (x^2+y^2)}=\left[1^2\cdot(+\infty+\infty)\cdot{1\over2}\right] =+\infty\)

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