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# ans0930 - Homework answers week of Sept 30#21 p 54(Theorem...

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Homework answers, week of Sept. 30 #21, p. 54 (Theorem 2.18.) If lim x a f ( x ) = L , L 6 = 0, and if g and k are such that g ( x ) 6 = 0 for 0 < | x - a | < k but lim x a g ( x ) = 0, then lim x a f ( x ) g ( x ) = . Discussion. In order to talk about the limit of f ( x ) / g ( x ) as x a , we must be assured that the ratio exists everywhere in some interval around a (except possibly at a itself). This requires that we not divide by zero, so g ( x ) cannot be zero in some interval around a . (An example is a polynomial with a root at a .) To establish a proof, we start with the deﬁnitions of the various limits and decide how to tie them together. In particular, since we have the freedom to choose the various epsilons associated with the limits of f and g , we show how to pick them so that the ratio | f ( x ) / g ( x ) | is as large as desired. Proof. We must show that, given B > 0, it is possible to ﬁnd δ such that ± ± ± ± f ( x ) g ( x ) ± ± ± ± > B whenever 0 < | x - a | < δ , (1) or, equivalently, | f ( x ) | > B | g ( x

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ans0930 - Homework answers week of Sept 30#21 p 54(Theorem...

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