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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 challengeresponse 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 &gt; 0 there exists &gt; 0 such that every x A with 0 &lt;  x c  &lt; satisfies f ( x ) &gt; M . Proof: Let M &gt; 0 be given. Take = 1 / M . Then 0 &lt;  x  &lt; = 1 / M implies 1 x 2 = 1  x  2 &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 &gt; 0 there exists M &gt; 0 such that every x A with x &gt; M satisfies  f ( x ) L  &lt; . Proof: Let &gt; 0 be given. Take M = 1 / . Then x &gt; implies 0 &lt; 1 /x &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 &gt; 0 there exists N &gt; such that every x A with x &gt; N satisfies f ( x ) &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 &gt; 0 be given. Using the continuity of g at f ( c ), choose &gt; 0 so that y B and  y f ( c )  &lt; implies  g ( y ) g ( f ( c ))  &lt; . Using the continuity of f at c , choose &gt; so that x A and  x c  &lt; implies  f ( x ) f ( c )  &lt; . Then, if x A and  x c  &lt; , one has  f ( x ) f ( c )  &lt; , hence (taking y = f ( x )) also  g ( f ( x )) g ( f ( c ))  &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.
 Fall '10
 Austin

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