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# hom4sol - Solutions to Homework 4 February 9 2009 20.10...

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Solutions to Homework 4 February 9, 2009 20.10 Prove that a: lim x →-∞ f ( x ) = , b: lim x 0 - f ( x ) = −∞ , c: lim x 0 + f ( x ) = , d: lim x 0 f ( x ) Does not exists, and e: lim x →∞ f ( x ) = −∞ for f ( x ) = 1 - x 2 x . You can do this by ǫ - δ or through the sequence definition. Proof: a: Let x n be a sequence in ( −∞ , 0) such that x n → −∞ , for any 1 M < 0 there is an integer N > 0 so that for all n > N implies x n < M + 1 M < 1 x n 1 /x n x n = f ( x n ) Since this is true for any sequens in the domain that diverges to infty lim x →-∞ f ( x ) = b: Let x n be a sequence in ( 1 , 0) such that x n 0, for any M > 0 there is an integer N > 0 so that for all n > N implies 0 > x n > 1 /M ⇒ − M > 1 /x n ⇒ − M + 1 > 1 /x n + 1 1 /x n x n = f ( x n ) Since this is true for any sequens in the domain that converge to 0 lim x →-∞ f ( x ) = −∞ c: Let x n be a sequence in (0 , 1) such that x n 0, for any M > 0 there is an integer N > 0 so that for all n > N implies 0 < x n < 1 /M M < 1 /x n M 1 > 1 /x n 1 1 /x n

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