Hw5sol - MT320 Homework 5 Solutions 3 April 2009 Exercise 4.2.7(a The statement lim x 1/x 2 = certainly makes intuitive sense Con struct a rigorous

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Unformatted text preview: MT320 Homework 5 Solutions 3 April, 2009 Exercise 4.2.7. (a) The statement lim x 1 /x 2 = certainly makes intuitive sense. Con- struct a rigorous definition in the challenge-response style of Definition 4.2.1 for a limit statement of the form lim x c f ( x ) = and use it to prove the previous statement. Answer: One has lim x c f ( x ) = + if and only if for every M > 0 there exists > 0 such that every x A with 0 < | x- c | < satisfies f ( x ) > M . Proof: Let M > 0 be given. Take = 1 / M . Then 0 < | x | < = 1 / M implies 1 x 2 = 1 | x | 2 > 1 (1 / M ) 2 = M, as desired. (b) Now, construct a definition for the statement lim x f ( x ) = L . Show lim x 1 /x = 0. Answer: One has lim x + f ( x ) = L if and only if for every > 0 there exists M > 0 such that every x A with x > M satisfies | f ( x )- L | < . Proof: Let > 0 be given. Take M = 1 / . Then x > implies 0 < 1 /x < 1 /M = , as desired. (c) What would a rigorous definition for lim x f ( x ) = look like? Give an example of such a limit. Answer: One has lim x + f ( x ) = + if and only if for every M > 0 there exists N > such that every x A with x > N satisfies f ( x ) > N . For example, f : R R with f ( x ) = x works: given M , take N = M . Exercise 4.3.2. (a) Supply a proof for Theorem 4.3.9 using the- characterization of continuity. Proof: Let > 0 be given. Using the continuity of g at f ( c ), choose > 0 so that y B and | y- f ( c ) | < implies | g ( y )- g ( f ( c )) | < . Using the continuity of f at c , choose > so that x A and | x- c | < implies | f ( x )- f ( c ) | < . Then, if x A and | x- c | < , one has | f ( x )- f ( c ) | < , hence (taking y = f ( x )) also | g ( f ( x ))- g ( f ( c )) | < , as desired....
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This note was uploaded on 11/22/2010 for the course MATH 115 taught by Professor Austin during the Fall '10 term at USC.

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Hw5sol - MT320 Homework 5 Solutions 3 April 2009 Exercise 4.2.7(a The statement lim x 1/x 2 = certainly makes intuitive sense Con struct a rigorous

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