Exercises5 - Exercises - 5 1- Evaluate the following...

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Exercises - 5 1- Evaluate the following limits, if exists. a) ( , ) (0,0) lim x y 2 2 2 2 xy x y + b) ( , ) (0,0) lim x y 3 2 6 4 2 xy x y + c) ( , ) (0,0) lim x y 2 2 xy x y + d) ( , ) (0,0) lim x y 3 3 2 x y x y + + e) ( , ) (0,0) lim x y 2 3 8( 2 ) xy x y + f) ( , ) (0,0) lim x y 2 2 2 2 x y x y - + g) ( , ) (0,0) lim x y 2 3 4 6 x y x y + h) ( , ) (1,3) lim x y 2 2 2 ( 1)( 3) ( 1) ( 3) x y x y - - - + - i) ( , ) (0,0) lim x y 2 2 2 1 ( )sin x y x y + + j) ( , ) (1,0) lim x y 3 2 2 ( 1) ( 1) x x y - - + 2) Given f(x,y) = y x , tan( ) x u v = , sin( ) y v u = , 2 u t = , ln v t = , find ( ) df dt π . 3) Let ( ) ( , ( , )) F x f x f x x = and 1 2 (1,1) 1, (1,1) , (1,1) f f a f b = = = . Find: a) (1) F b) (1) dF dx 4) Let 2 ( , ) xy z f xy e = and assume the following: 2 2 2 11 11 12 21 (2, ) 1, (2,1) 2, (2, ) (2, ) 0 f e f f e f e = = = = 2 21 12 22 22 (2,1) (2,1) 1, (2, ) 1, (2,1) 2 f f f e f = = = = 2 2 1 1 2 2 (2, ) 0, (2,1) 1, (2, ) 2, (2,1) 4 f
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