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# hw4 - R be a continuous function If lim x →∞ f x exists...

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Math 125A, Fall 2009 Homework 4 Due Date: October 23, 2009 Problem 1: Find the limit lim x 0 x cos ( 3 x 2 + e sin x - ln | x | ) . and justify your answers. Problem 2: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If f : R R is a continuous function, and if the sequence a n = f ( n ) converges to L , then lim x →∞ f ( x ) = L also. (b) If lim x →∞ f ( x ) = and lim x →∞ g ( x ) = -∞ , then lim x →∞ [ f ( x ) + g ( x )] does not exist. (c) If lim x →∞ f ( x ) = and lim x →∞ g ( x ) = 0, then lim x →∞ f ( x ) g ( x ) does not exist. Problem 3: Let f : ( a, b ) R be any function, and assume x 0 ( a, b ). Consider the following two limit statements: (i) lim h 0 [ f ( x 0 + h ) - f ( x 0 )] = 0 (ii) lim h 0 [ f ( x 0 + h ) - f ( x 0 - h )] = 0 (a) Prove that (i) implies (ii). (b) Give an example of a function for which (ii) holds but (i) does not. Problem 4: Let f : [0 ,
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Unformatted text preview: R be a continuous function. If lim x →∞ f ( x ) exists and is ﬁnite, then prove that f is actually uniformly continuous. (Note that Theorem 19.2 does not apply because the domain of f is not bounded.) Problem 5: Determine the radius of convergence of the series ∑ ∞ n =0 a n x n , where: a n = 5 n , if n is odd 1 2 n , if n is even Problem 6: Deﬁne functions f n : R → R by f n ( x ) = cos n x . Note that these functions are all continuous. Prove that: (a) lim n →∞ f n ( x ) = 0 if x is not a multiple of π . (b) lim n →∞ f n ( x ) = 1 if x is an even multiple of π . (c) lim n →∞ f n ( x ) does not exist if x is an odd multiple of π ....
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