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Quiz 4 Solution - MAC2313 Quiz 4 Solution 1 Find the limit...

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MAC2313 Quiz 4 Solution 1. Find the limit, if it exists, or show that the limit does not exist. (a) lim ( x,y ) (0 , 0) 2 xy x 2 + y 2 Solution: Along any linear approach y = mx the limit is lim ( x,mx ) (0 , 0) 2 mx 2 (1 + m 2 ) x 2 = 2 m 1 + m 2 . Therefore, an approach with slope 0 has limit 0 and an approach with slope 1 has limit 1. Hence, the limit does not exist. (b) lim ( x,y ) (0 , 0) x 3 - y 3 x 2 + y 2 Solution: Transforming into polar coordinates we find lim ( x,y ) (0 , 0) x 3 - y 3 x 2 + y 2 = lim r 0 + r 3 (cos 3 ( θ ) - sin 3 ( θ )) r 2 = lim r 0 + r (cos 3 ( θ ) - sin 3 ( θ )) . Since - 2 r r (cos 3 ( θ ) - sin 3 ( θ )) 2 r and lim r 0 + - 2 r = 0 = lim r 0 + 2 r, the Squeeze Theorem gives lim r 0 + r (cos 3 ( θ ) - sin 3 ( θ )) = 0 . 2. Let f ( x, y ) = e x 2 - y 2 . First show that f is differentiable, then determine the linearization about the point (1 , - 1). Solution: Taking the partial derivatives f x ( x, y ) = 2 xe x 2 - y 2 f y ( x, y ) = - 2 ye x 2 - y 2 Both f x and f y are continuous and exist at (1 , - 1), therefore the function is differentiable.
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