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Unformatted text preview: MAC2313 Exam 2 Dr Sin No Calculators. Answer the questions in the spaces provided on the question sheets. Please write your answers in full detail. If you run out of room for an answer, continue on the back of the page or on other paper. Name: 1. (7 points) Let f ( x,y ) = ( yx 2 x 4 + y 2 if ( x,y ) 6 = (0 , 0) if ( x,y ) = (0 , 0) . (a) (1 point) At which points of the plane is the function f ( x,y ) defined? (b) (4 points) At which points of the plane does the function f ( x,y ) have a limit? (c) (2 points) At which points of the plane is the function f ( x,y ) continuous? Solution: (a)The function is defined at all points of the plane. (b)At every point ( x,y ) 6 = (0 , 0) the function f ( x,y ) is defined by a rational function (whose denominator is not zero at ( x,y )), so f ( x,y ) has a limit there. At (0 , 0), we have lim x x 4 +0 = 0 along x = 0, whereas lim x x 4 x 4 + x 4 = 1 / 2 along y = x 2 , so f ( x,y ) has no limit at (0 , 0)....
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- Spring '08