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# ISM_T11_C02_B - Section 2.5 Infinite Limits and Vertical...

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Section 2.5 Infinite Limits and Vertical Asymptotes 93 (d) lim lim xx Ä±" x 1 (x 1)(x 1) negative negative negative # ±² ± œœ ± _ Š‹ 19. (a) lim 0 Ä! x x x negative # "" " ±œ² œ ± _ (b) 0 x x x positive # " œ_ (c) lim 2 2 0 x2 Ä \$ È x ## Î\$ "Î\$ # ±"Î\$ ±"Î\$ ± œ ± œ±œ (d) x1 Ä± x3 # ± # " ±œ± œ ˆ‰ 20. (a) lim (b) lim x x Ä±# 2x 4 positive 2x 4 negative positive positive # # ±± ²² œ±_ (c) 0 Ä" 2 0 2x 4 2x 4 4 (x 1)(x 1) # ± # ² ²± œ (d) x 2x 4 4 # " ² œ 21. (a) x 2x x (x 2) positive negative (x 2)(x 1) negative negative # \$# # ± _ (b) , x 2 x Ä# x 3 x x ( ) x 4 (x 2)(x 1) # # # ± " œ Á (c) , x 2 x x x ( ) x 4 (x 2)(x 1) # # # ± " œ Á (d) , x 2 x x x ( ) x 4 (x 2)(x 1) # # # ± " œ Á (e) x 2x x (x 2) positive negative (x 2)(x 1) negative negative # # ± _ 22. (a) x x 4x x(x )(x 2) x(x ) (4) 8 (x 2)(x ) (x 1) # \$ " " # ² ² # # " ± œ œ (b) lim lim lim x x 4x x(x )(x 2) x(x ) negative positive (x 2)(x ) (x 1) negative # \$ # ² ² # " ± œ _ (c) x0 Ä x 4x x(x )(x 2) x(x ) negative positive (x 2)(x ) (x 1) negative # \$ # ² ² # " ± œ _ (d) 0 x 0 x 4x x(x )(x 2) x(x ) (1)(3) (x 2)(x ) (x 1) # \$ # ² ² # " ± œ œ (e) x x x(x ) positive positive negative ±" ²# and x x x(x ) negative positive negative so the function has no limit as x 0. Ä 23. (a) 2 (b) 2 t t ± ± ±œ ± _ _ 3 3 t t "Î\$ "Î\$ 24. (a) 7 (b) 7 t t ± ± " " t t \$Î& \$Î& ²œ _ ± _ 25. (a) (b) x x ’“ " " x( x 1 ) x 1 ) 2 2 # Î\$ #Î\$ #Î\$ #Î\$ _ _ (c) (d) x x " " x 1 ) x 1 ) 2 2 # Î\$ #Î\$ #Î\$ #Î\$ _ _ 26. (a) (b) x x " " x 1 ) x 1 ) 1 1 "Î\$ %Î\$ "Î\$ %Î\$ _ ± _ (c) (d) x x " " x 1 ) x 1 ) 1 1 "Î\$ %Î\$ "Î\$ %Î\$ ± _ ± _

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94 Chapter 2 Limits and Continuity 27. y 28. y œœ " " ± ² x1 29. y 30. y " ± ± x4 x3 3 31. y 1 32. y ± # ² 2x 2 x2 x ²" ²² # 33. y
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ISM_T11_C02_B - Section 2.5 Infinite Limits and Vertical...

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