limit_warning

limit_warning - A small warning about taking limits Math 8...

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A small warning about taking limits Math 8 2009, Chris Lyons Consider the following evaluation of a limit: lim k →∞ 1 k 2 = lim k →∞ 1 k · 1 k = ± lim k →∞ 1 k ² · ± lim k →∞ 1 k ² = 0 · 0 = 0 . (1) Here’s the basic principle that we used in this calculation: lim k →∞ a n b n = ± lim k →∞ a n ² · ± lim k →∞ b n ² (2) Q : How correct is this principle? Before answering this question, let’s look at two more examples which make us think that there’s a little more going on in (2) than first meets the eye: 1. Consider 1 = lim k →∞ 1 = lim k →∞ ( - 1) 2 k = lim k →∞ [( - 1) k · ( - 1) k ] = ± lim k →∞ ( - 1) k ² · ± lim k →∞ ( - 1) k ² . But this expression doesn’t make sense because the sequence ³ ( - 1) k ´ k =1 doesn’t have a limit! 2. Consider 1 = lim k →∞ 1 = lim k →∞ k · 1 k = ± lim k →∞ k ² · ± lim k →∞ 1 k ² = ± lim k →∞ k ² · 0 . This expression doesn’t make sense either, since the sequence { k } k =1 doesn’t have a limit. These examples should help us see what kind of care we need to take when using principle (2). Namely,
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