hwsol_13

hwsol_13 - Homework Assignment 13 - (12.2) - Solutions Page...

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Homework Assignment 13 - (12.2) - Solutions Page 934: 6*, 10*, 14*, 16*, 18*, 26*, 30*, 32*, 42*, 44*, 48*, 55*, 58* 6. As ! x , y " ! ! 2, ! 1 " , f ! x , y " ! 2 " ! ! 1 " 2 2 ! 2 ! 2 "! ! 1 " # 1 8 lim ! x , y " ! ! 2, ! 1 " x " y x 2 ! 2 xy # 1 8 10. As ! x , y , z " ! ! 1,1,2 " , f ! x , y , z " ! e 1 " 1 ! 2 1 ! 2 # ! 1 lim ! x , y " ! ! 1,1,2 " e x " y ! z x ! z # ! 1 14. As ! x , y " ! ! 0,0 " , f ! x , y " ! 0 0 Along y # 0, lim ! x , y " ! ! 0,0 " 2 xy x 2 " 2 y 2 # lim x ! 0 0 x 2 " 0 # 0 Along y # x , lim ! x , y " ! ! 0,0 " 2 xy x 2 " 2 y 2 # lim x ! 0 2 x 2 x 2 " 2 x 2 # lim x ! 0 2 3 # 2 3 lim ! x , y " ! ! 0,0 " 2 xy x 2 " 2 y 2 DNE 16. As ! x , y " ! ! " , f ! x , y " ! 0 0 Along y # 0, lim ! x , y " ! ! 0,0 " 3 x 3 y x 4 " y 2 # lim ! x , y " ! ! 0,0 " 0 x 4 " 0 # 0 Along y # x , lim ! x , y " ! ! 0,0 " 3 x 3 y x 4 " y 2 # lim x ! 0 3 x 4 x 2 " x 4 # 3 2 So, lim ! x , y " ! ! 0,0 " 3 x 3 y x 4 " y 2 DNE 18. Along y # 0, lim ! x , y " ! ! 0,0 " 2 xy 3 x 2 " 8 y 6 # lim x ! 0 0 x 2 " 0 # 0 Along y 3 # x , lim ! x , y " ! ! 0,0 " 2 xy 3 x 2 " 8 y 6 # lim x ! 0 2 x 2 x 2 " 8 x 3 # 2 9 So, lim ! x , y " ! ! 0,0 " 2 xy 3 x 2 " 8 y 6 DNE 26. Along x ! axis ( y # 0, z # 0 " , lim ! x , y , z " ! ! 0,0,0 " x 2 yz x 4 " y 4 " z 4 # lim x ! 0 0 x 4 " 0 " 0 # 0 1
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Along the line an intersection of planes: y # x and z # x : lim ! x , y , z " ! ! 0,0,0 " x 2 yz x 4 " y 4 " z 4 # lim x ! 0 x 4 x 4 " x 4 " x 4 # 1 3 lim ! x , y , z " ! ! 0,0,0 " x 2 yz x 4 " y 4 " z 4 DNE 30. lim ! x , y " ! ! 0,0 " x 3 y " x 2 y 3 x 2 " y 2 # lim r ! 0 r 4 cos 3 ! sin ! " r 5 cos 2 ! sin 3 ! r 2 # lim r ! 0 r 2 cos 3 ! sin ! " r 3 cos 2 ! sin 3 ! # 0 32. lim ! x , y " ! ! 0,0 " x 2 y ! x 2 ! y 2 x 2 " y 2 # lim ! x , y " ! ! 0,0 " x 2 y x 2 " y 2 ! 1 # lim r ! 0 r 3 cos 2 ! sin ! r 2 ! 1 # lim r ! 0 ! r cos 2 ! sin ! ! 1 " # ! 1 42. f ! x , y " # x 2 ! y 2 ! 1 f is continuous if x 2 ! y 2 ! 1 " 0, that x 2 ! y 2 " 1. So, f is continuous on D # ! x , y " ; x 2 ! y 2 " 1 44. f ! x , y " # tan ! x " y " f is continuous if x " y # n " 2 , where n #$ 1, $ 3, $ 5, . .. So, f is continuous on D # ! x , y " ; x " y # n " 2 , wheren 1, $ 3, $ 48. f ! x , y , z " # z ! x 2 ! y 2 f is continous if z ! x 2 ! y 2 " 0, that is, x 2 " y 2 $ z . So, f is continuous on D # ! x , y , z " ; x 2 " y 2 $ z 55. f ! x , y " # xy 2 x 2 " y 4 if ! x , y " # ! 0,0 " 0i f ! x , y " # ! " Show f ! x , y " is not continous at ! " by showing ! 1 " lim ! x , y " ! ! 0,0 " xy 2 x 2 " y 4 DNE or ! 2 " lim ! x , y " ! ! 0,0 " xy 2 x 2 " y 4 # 0. Along y # 0, lim ! x , y " ! ! 0,0 " xy 2 x 2 " y 4 # lim x ! 0 0 x 2 " 0 # 0 Along y # x , lim ! x , y " ! ! 0,0 " xy 2 x 2 " y 4 # lim x ! 0 x 2 x 2 " x 2 # 1 2 So, lim ! x , y " ! ! 0,0 " xy 2 x 2 " y 4 DNE and therefore, f ! x , y " is not continuous at ! " .
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hwsol_13 - Homework Assignment 13 - (12.2) - Solutions Page...

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