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quiz_02A - r → ± 2 r = 0 it follows from the Squeeze...

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Math 213 — Quiz 2 Name 1. Use polar coordinates to evaluate lim ( x,y ) (0 , 0) x 3 + y 3 x 2 + y 2 . Using x = r cos θ , y = r sin θ , and r 2 = x 2 + y 2 , we find x 3 + y 3 x 2 + y 2 = r 3 cos 3 θ + r 3 sin 3 θ r 2 = r (cos 3 θ + sin 3 θ ) . Now, regardless of the value of θ , we have | cos 3 θ + sin 3 θ | ≤ 2. Hence, - 2 r r (cos 3 θ + sin 3 θ ) +2 r and since lim
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Unformatted text preview: r → + ± 2 r = 0, it follows from the Squeeze Theorem that lim r → + r (cos 3 θ + sin 3 θ ) = 0 . We conclude that lim ( x,y ) → (0 , 0) x 3 + y 3 x 2 + y 2 = 0 ....
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