A2_soln - Math 237 Assignment 2 Due Friday Sept 26th sin(xy...

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Math 237 Assignment 2 Due: Friday, Sept 26th 1. Determine if f is continuous at (0 , 0) where f ( x,y ) = ( sin( xy ) ln( x 2 + y 2 +1) if ( x,y ) 6 = (0 , 0) 0 if ( x,y ) = (0 , 0) . Solution: f ( x,y ) is not continuous at (0 , 0), since if we approach the limit along y = x we get (using Taylor series) lim ( x,y ) (0 , 0) f ( x,y ) = lim x 0 sin x 2 ln(2 x 2 + 1) = lim x 0 x 2 - x 6 6 + ··· 2 x 2 - (2 x 2 ) 2 2 + ··· = lim x 0 1 - x 4 6 + ··· 2 - 2 x 2 + ··· = 1 2 , (L’Hospital’s rule would also work), which is not equal to f (0 , 0), thus f is not continuous at (0 , 0). 2. Determine where the function f ( x,y ) = p x 2 - y 2 is continuous. Justify your answer. Solution: From assignment 1, we know that f ( x,y ) is not defined at every point in any neighborhood of points for which x 2 - y 2 = 0. Hence, by definition, f ( x,y ) can not be continuous there. For any point ( x,y ) satisfying x 2 - y 2 > 0, we have that f ( x,y ) is continuous by the continuity theorems. 3.
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This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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A2_soln - Math 237 Assignment 2 Due Friday Sept 26th sin(xy...

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