quiz1-sol - ≤ x 2 y 2 x 6 + y 2 = x 2 y 2 x 6 + y 2 ≤ x...

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Date: June 12, 2009, Friday NAME:. .................................................... STUDENT NO:. .................................................... SECTION NUMBER : ..................................................... Math 116 Calculus – QUIZ # 1 Q-1) Find the following limits, if they exist, and prove your results: (i) lim ( x,y ) (0 , 0) x 2 y 2 x 6 + y 2 , (ii) lim ( x,y ) (0 , 0) x 2 y x 6 + y 2 . Solutions: For the first one observe that
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Unformatted text preview: ≤ x 2 y 2 x 6 + y 2 = x 2 y 2 x 6 + y 2 ≤ x 2 , and by the sandwich theorem, the limit is zero. For the second one, try the path y = λx 2 : lim ( x,y ) → (0 , 0) y = λx 2 x 2 y x 6 + y 2 = lim x → λx 4 x 6 + λ 2 x 4 = lim x → λ x 2 + λ 2 = 1 λ · The limit on different paths do not agree so the limit does not exist....
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