Limits and Continuity

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

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Unformatted text preview: Contents 2 Limits and Continuity 35 2.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Numerical Introduction to Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 CONTENTS 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 → ( | x | - | - 3 | ) (f) lim x → a ( | x | - | a | ) (g) lim z →- 2 ( | z | 2 + 2 | z | - 3) (h) lim x →- 2 x | x | 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 → 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 → x 2- x x (c) lim h → 4( x + h ) 3- 4 x 3 h (d) lim x → 1- 2 2 x 1- 2 x (e) lim x → 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 → 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 → + f ( x...
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Limits and Continuity - Contents 2 Limits and Continuity 35...

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