hw5sol

# 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 &amp;gt; 0 there exists &amp;gt; 0 such that every x A with 0 &amp;lt; | x- c | &amp;lt; satisfies f ( x ) &amp;gt; M . Proof: Let M &amp;gt; 0 be given. Take = 1 / M . Then 0 &amp;lt; | x | &amp;lt; = 1 / M implies 1 x 2 = 1 | x | 2 &amp;gt; 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 &amp;gt; 0 there exists M &amp;gt; 0 such that every x A with x &amp;gt; M satisfies | f ( x )- L | &amp;lt; . Proof: Let &amp;gt; 0 be given. Take M = 1 / . Then x &amp;gt; implies 0 &amp;lt; 1 /x &amp;lt; 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 &amp;gt; 0 there exists N &amp;gt; such that every x A with x &amp;gt; N satisfies f ( x ) &amp;gt; 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 &amp;gt; 0 be given. Using the continuity of g at f ( c ), choose &amp;gt; 0 so that y B and | y- f ( c ) | &amp;lt; implies | g ( y )- g ( f ( c )) | &amp;lt; . Using the continuity of f at c , choose &amp;gt; so that x A and | x- c | &amp;lt; implies | f ( x )- f ( c ) | &amp;lt; . Then, if x A and | x- c | &amp;lt; , one has | f ( x )- f ( c ) | &amp;lt; , hence (taking y = f ( x )) also | g ( f ( x ))- g ( f ( c )) | &amp;lt; , 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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