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Limits and Continuity - Contents 2 Limits and Continuity 35...

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Contents 2 Limits and Continuity 35 2.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Numerical Introduction to Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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2 CONTENTS
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Chapter 2 Limits and Continuity 2.1 Limits 1. Define f : R R by the rule f ( x ) = 101 for every x in the domain of f . Find the following limits if they exist: (a) lim x 101 f ( x ) (b) lim x →- 100 f ( x ) (c) lim x 20 f ( x ) 101 - f ( x ) (d) lim x →- 1 [ f ( x )] 2 - 2 f ( x ) 2. Let g : R R be defined by the rule g ( x ) = x 2 - 4 for all x in the domain of g and L : R R be defined by the rule L ( x ) = 2 x - 1 for all x in the domain of L . Find the following limits: (a) lim x 2 g ( x ) (b) lim x →- 2 g ( x ) (c) lim x 3 g ( x ) - L ( x ) (d) lim x 1 [ g ( x ) · L ( x )] (e) lim x 1 / 2 g ( x ) L ( x ) (f) lim x →- 1 / 2 g ( x ) L ( x ) (g) lim x 2 [ g ( x ) - g (2)] (h) lim x 1 [ g ( x ) - g ( L (0))] 3. Evaluate the following limits: (a) lim y 3 | y | (b) lim w →- 2 | w | (c) lim t 3 | t - 2 | (d) lim y 61 | y | | y | (e) lim x 0 ( | x | - | - 3 | ) (f) lim x a ( | x | - | a | ) (g) lim z →- 2 ( | z | 2 + 2 | z | - 3) (h) lim x →- 2 x | x |
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36 Limits and Continuity 4. Evaluate the following limits: (a) lim u 3 2 u 2 - 4 u - 2 (b) lim u 2 u 2 - 4 u - 2 (c) lim v 0 v 2 + v - 2 v +2 (d) lim v →- 2 v 2 + v - 2 v +2 (e) lim x π 4 sin 2 x - cos 2 ( x ) sin x +cos x (f) lim x 17 (sin 2 x + cos 2 x ) (g) lim x π 2 sin x 2 +cos x 2 sin x 2 5. Find the following limits if they exist: (a) lim x 2 x - 2 x - 2 (b) lim x 3 x 2 - 9 x - 3 (c) lim x a 1 /x - 1 /a x - 1 , a 6 = 0 (d) lim x 4 | x |- 4 x - 4 (e) lim x 1 2 x 3 - 3 x 2 +2 x - 1 x - 1 (f) lim x →- 3 x 3 +27 x +3 6. Find the following limits if they exist: (a) lim x a x 4 - a 4 x - a (b) lim x 0 x 2 - x x (c) lim h 0 4( x + h ) 3 - 4 x 3 h (d) lim x 0 1 - 2 2 x 1 - 2 x (e) lim x 0 1 - 2 2 x 1+2 x (f) lim x 10 (1 - log 10 x ) 7. Find lim x a f ( x ) - f ( a ) x - a and lim t 0 f ( a + t ) - f ( a ) t for: (a) f ( x ) = x 2 , a = 3 (b) f ( x ) = x 2 , a = - 1 / 3 (b) f ( x ) = x 2 + 1 , a = 2 (d) f ( x ) = 3 x 2 - x, a = 0 (e) f ( x ) = 1 / 2 x 2 - 3 x + 1 , a = 0 (f) f ( x ) = ( x - 3) 2 - 5 , a = 1 (g) f ( x ) = | x | 2 , a = 2 (h) f ( x ) = x | x | , a = - 2 In problems 8 to 13 find lim x a g ( x ) - g ( a ) x - a for the given function g and the value a . 8. g ( x ) = x + 2 , a = 93 9. g ( x ) = - 2 x, a = 14 10. g ( x ) = x (1 - x ) , a = 1 11. g ( x ) = ( x + 1) 2 , a = - 1 12. g ( x ) = 2 x 2 + 1 , a = 2 13. g ( x ) = 1 x +1 , a = 1 Find the following limits if they exist: 14. lim x 1 + | x - 1 | x - 1 and lim x 1 - | x - 1 | x - 1 15. lim x →- 2 + x +2 | x +2 | and lim x →- 2 - x +2 | x +2 | 16. lim x 0 + f ( x ), where f ( x ) = x 2 , x < 0 x + 1 , x 0
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2.1 Limits 37 17. lim x 0 - f ( x ), where f ( x ) is defined in question 16. 18. lim x 1 + g ( x ), where g ( x ) = ( x - 1) 2 , x > 1 ( x - 1) 3 , x < 1 19. lim x 1 - g ( x ), where g ( x ) is defined in question 18.
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