Limit Review Problem

Limit Review Problem - y = mx we get lim x f ( x, mx ) =...

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MATH150 Limit Review Problem 12/2/05 Problem : For each of the following limits, compute its value or show that it doesn’t exist. (a) lim ( x,y ) (0 , 0) tan( x 2 + y 4 ) x 2 + y 4 . (b) lim ( x,y ) (0 , 0) xy 2 p x 4 + y 8 . Solution : (a) This is a one-dimensional limit “in disguise”: let t = x 2 + y 4 . Then lim ( x,y ) (0 , 0) tan( x 2 + y 4 ) x 2 + y 4 = lim t 0 tan t t l’Hˆop = lim t 0 sec 2 t 1 = 1 (b) Let f ( x, y ) = xy 2 x 4 + y 8 . If we approach the origin along the line x = 0 we get lim y 0 f (0 , y ) = lim y 0 0 = 0. If we approach the origin along the line y = 0 we get lim x 0 f ( x, 0) = lim x 0 0 = 0. If we approach the origin along the line
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Unformatted text preview: y = mx we get lim x f ( x, mx ) = lim x m 2 x 3 x 4 + m 8 x 8 = lim x m 2 x 3 x 2 1 + m 8 x 4 = lim x m 2 x 1 + m 8 x 4 = 0 However, if we approach the origin along the parabola x = y 2 we get lim y y 4 p 2 y 8 = lim y y 4 y 4 2 = 1 2 Thus lim ( x,y ) (0 , 0) xy 2 p x 4 + y 8 doesnt exist....
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