a5 solu - MATH 251-3, Fall 2007 Simon Fraser University...

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Unformatted text preview: MATH 251-3, Fall 2007 Simon Fraser University Assignment 5: Solutions Additional Question Due: 4:30pm, Wednesday 24 October 2007 1. (a) Find the limit, if it exists, or show carefully that the limit does not exist: lim ( x,y ) → (0 , 0) 3 xy (2- cos( x )) x 2 + 2 y 2 Solution: We can show that the limit does not exist by taking the limit lim ( x,y ) → (0 , 0) f ( x, y ) = lim ( x,y ) → (0 , 0) 3 xy (2- cos( x )) x 2 + 2 y 2 along two different paths, and showing that they are different. For instance, along the x-axis y = 0 , we have f ( x, 0) = 3 x · 0(2- cos( x )) x 2 + 0 = 0 = ⇒ lim ( x,y ) → (0 , 0) f ( x, y ) = 0 along y = 0 (similarly, the limit along the y-axis x = 0 is 0). However, along the line y = x , f ( x, x ) = 3 x · x (2- cos( x )) x 2 + 2 x 2 = 2- cos( x ) , so the limit along this line is lim ( x,y ) → (0 , 0) f ( x, y ) = lim x → (2- cos( x )) = 2- 1 = 1 6 = 0 along y = x. Since the limits along different paths are different, the limit lim ( x,y ) → (0 , 0) f ( x, y ) does not exist. [In general, along the straight line y = mx we have f ( x, mx ) = 3 x · mx (2- cos( x )) x 2 + 2( mx ) 2 = 3 m 1 + 2 m 2 (2- cos( x )) = ⇒ lim ( x,y ) → (0 , 0) f ( x, y ) = 3 m 1 + 2 m 2 along y = mx....
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This note was uploaded on 10/09/2009 for the course MATH macm 101 taught by Professor Jcliu during the Spring '09 term at Simon Fraser.

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a5 solu - MATH 251-3, Fall 2007 Simon Fraser University...

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