Chapter 14 B - Section 14.2 Limits and Continuity 6. x y 0...

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Section 14.2 Limits and Continuity 871 6. lim cos cos cos 0 1 ÐßÑÄÐßÑ xy 00 Š‹Š‹ xy1 001 #$ ± ±± ± œœ œ 7. lim e e e ÐßÑÄÐß Ñ 0ln2 0 l n 2 l n ²² " # œ ˆ‰ 1 2 8. lim ln 1 x y ln 1 (1) (1) ln 2 11 kk k k ± œ ± œ ## # # 9. lim lim e e lim 1 1 1 Ä x0 e sin x sin x sin x xx x y y œ œ ab ! †† 10. lim cos xy 1 cos (1)(1) 1 cos 0 1 ˆ È È $$ ² œ ² 11. lim 0 10 x sin y x1 11 2 1s in 0 0 œ 12. lim 2 ÐßÑÄ ß 0 1 2 cos y 1 (cos 0) ys i n x 1 0s i n " ² ± œ ² 1 # 13. lim lim lim (x y) ( 1) 0 Á x 2xy y (x y) # ²± ² ² œ"² œ 14. lim lim lim (x y) (1 1) 2 Á x y (x y)(x y) ² ± œ ± œ 15. lim lim lim (y 2) (1 2) 1 ÁÁ x1 x yy2 x2 ( ) ( y2 ) ²² ± ² ² ² œ ² œ ² 16. lim lim lim ÐßÑÄÐß± ÑÐ ß Ñ Ä Ð ß ± ß Ñ Ä Ð ß ± Ñ Á ± ± 2 4 y 4 , , # y4 x y xy 4x 4x x(x 1)(y 4) x(x 1) 1 ²± ² ² ± ² # " œ (2 1) ²# " œ 17. lim lim lim x y 2 xy2x2y x y x ² ² ± ± ÈÈ È È ˆ È È 2 2 œ œ Š‹ Note: (x y) must approach (0 0) through the first quadrant only with x y. ßß Á 18. lim lim lim x y 2 ² Á ² Á ² Á 22 xy4 xy2 ±² È 222 224 œ œ ± œ È 19. lim lim lim ± Á ± Á 20 2x y 4 2x y 4 È 2 2 2 2 2 2 # " œ "" " # ± È (2)(2) 0 4 20. lim lim lim ± Á ± Á 43 È È 1 1 1xy 1 1 "
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872 Chapter 14 Partial Derivatives œœ œ "" " ±± ± È È 43 1 22 4 21. lim ÐßßÑ 134 Š‹ """ ± ± xyz 1 2 1 2 1 243 1 9 œ 22. lim TÄÐß± ß± Ñ 111 2xy yz 2(1)( 1) ( 1)( 1) xz 1( 1 ) 11 2 ±² ± ² ² ²± # " " ## # # œ ² 23. lim sin x cos y sec z sin 3 cos 3 sec 0 1 1 2 330 ab a b ### # # œ œ ± œ 24. lim tan (xyz) tan 2 tan ±ßß ˆ‰ 1 42 1 2 ²" ²" ²" " # œ ² œ ² ˆ 44 †† 25. lim ze cos 2x 3e cos 2 (3)(1)(1) 3 1 03 Ð Ñ 2y 2 0 œ 1 26. lim ln x y z ln 0 ( 2) 0 ln 4 ln 2 TÄÐß±ßÑ 02 0 ÈÈ È # œ ±² ± 27. (a) All (x y) ß (b) All (x y) except (0 0) ßß 28. (a) All (x y) so that x y ßÁ (b) All (x y) ß 29. (a) All (x y) except where x 0 or y 0 ßœ œ (b) All (x y) ß 30. (a) All (x y) so that x 3x 2 0 (x 2)( 1) 0 x 2 and x 1 ß ²±ÁÊ ² B ² ÁÊÁ Á # (b) All (x y) so that y x # 31. (a) All (x y z) (b) All (x y z) except the interior of the cylinder x y 1 ± œ 32. (a) All (x y z) so that xyz 0 ³ (b) All (x y z) 33. (a) All (x y z) with z 0 Á (b) All (x y z) with x z 1 ± Á 34. (a) All (x y z) except (x 0 0) (b) All (x y z) except ( y 0) or (x 0 0) ß ß !ß ß ß ß 35. lim lim ; ÐßÑÄÐßÑ œ ² ÄÄ Ä Ä xy 00 along y x x0 ² œ ² œ ² œ ² œ ² œ ² xx x x 2 x 2 x 2 2 ÈÈÈ È È È kk ± ± lim œ ³ Ä along y x ² œ ² œ ² xxx 2 x 2( x) 2 2 È È È ± ²
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Section 14.2 Limits and Continuity 873 36. lim lim 1; lim ÐßÑÄÐßÑ œ ÄÄ Ä œ xy 00 along y 0 x0 along y x xx x 2 x %% % %# % # ±± ± ± œœ œ œ œ # ab " # 37. lim different limits for d œ along y kx # xk x x x x 1k % # # % # # ² ² ± ²² ± œ Ê ifferent values of k 38. lim ; if k 0, the limit is 1; but if k œ Á Ä along y kx k0 xy x(kx) xy x(kx) kx k kx k kk k k k k k k œ ± # # ²³ 0, the limit is 1 39. lim different limits for different values of k, k 1 œ Á ± Ä along y kx k1 x x ² ± Ê Á ³ 40. lim different limits for different values of k, k 1 œ Á Ä along y kx x x ± ² Ê Á 41. lim different limits for different values of k, k 0 œ Á Ä along y kx # yk x k x # ## # ± Ê Á 42. lim
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This note was uploaded on 02/16/2008 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).

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Chapter 14 B - Section 14.2 Limits and Continuity 6. x y 0...

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