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Unformatted text preview: Mathematics 75 Limit Proofs October 25, 2006 Every time you’re asked to prove that lim x → a f ( x ) = L , your answer will follow the same general form. How you fill in some of the details may vary, but the framework remains the same each time. An example of this frame work is: Prove that lim x → a f ( x ) = L . Proof. Let ε > 0 be given. Choose δ = . If 0 < | x- a | < δ , then | f ( x )- L | = = (You’ll have to fill in some work) . . . < ε. So the proof essentially has two steps. First, you have to choose a δ . Then, assume that 0 < | x- a | < δ and derive the inequality | f ( x )- L | < ε . Now seems like a good time for an example. Prove that lim x → 4 ( 1 2 x- 3) =- 1. Proof. Let ε > 0 be given. Now I’ll do a little computation to determine a good choice of δ . This is the kind of work I would normally do on a piece of scratch paper or a separate blackboard. Remember that our final goal is 1 2 x- 3- (- 1) < ε . That inequality is the same as 1 2 x- 2 < ε . If I factor a....
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This note was uploaded on 02/28/2008 for the course MATH 75 - 76 taught by Professor Yukich during the Spring '06 term at Lehigh University .
- Spring '06