2ndsol - Math 119 Week 2 Solutions 1 1. Guessing δ For...

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Unformatted text preview: Math 119 Week 2 Solutions 1 1. Guessing δ For given > 0, we want to find δ > 0 such that | (2 x + 3)- 5 | < whenever | x- 1 | < δ ⇒ | 2( x- 1) < whenever | x- 1 | < δ ⇒ | x- 1 | < 2 whenever | x- 1 | < δ Thus, we can choose δ = 2 Showing that this δ works ∀ > ∃ δ = 2 such that | (2 x + 3)- 5 | < whenever | x- 1 | < δ = 2 | (2 x + 3)- 5 | = | 2 x- 2 | = 2 | x- 1 | < 2 . 2 = 2. Guessing δ For given M , we want to find δ > 0 such that | 1 ( x + 3) 4 | > M whenever | x- (- 3) | < δ ⇒ | ( x + 3) 4 | < 1 M whenever | x + 3 | < δ ⇒ | ( x + 3) | < 4 r 1 M whenever | x + 3 | < δ so, we can choose δ = 4 r 1 M Showing that this δ works ∀ > ∃ δ = 4 r 1 M such that | 1 ( x + 3) 4 | > M whenever | x- (- 3) | < δ = 4 q 1 M | 1 ( x + 3) 4 | > 1 δ 4 = 1 ( 4 q 1 M ) 4 = M 3. We want that f ( x ) is continuous on (-∞ , ∞ ). Obviously f ( x ) is continu- ous on (-∞ , 3) and (3 , ∞ ), because f ( x ) is defined as a polinomial on the ray (-∞ , 3) or (3 , ∞ ). Thus, it is enough that lim)....
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This note was uploaded on 03/31/2010 for the course MATHEMATIC 119 taught by Professor Muhiddinuğuz during the Fall '08 term at Middle East Technical University.

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2ndsol - Math 119 Week 2 Solutions 1 1. Guessing δ For...

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