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Topic: Limit of Product of Functions (Read 258 times) |
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william wu
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Limit of Product of Functions
« on: Mar 10th, 2004, 10:29am » |
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Suppose lim x[to]c f(x) = a, and lim x[to]c g(x) = b.
Is it necessarily true that lim x[to]c f(x)g(x) = ab ?
Offer a counterexample if false, or prove it if true.
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| « Last Edit: Mar 10th, 2004, 10:29am by william wu » |
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kellys
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Re: Limit of Product of Functions
« Reply #1 on: Mar 10th, 2004, 12:15pm » |
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[e]Assuming a,b,c are not infinity,[/e]
Use fact that,
|fg(x)-ab| = |fg(x)-ag(x)-bf(x)+ab + ag(x)-ab + bf(x)-ab|
and,
|fg(x)-ag(x)-bf(x)+ab|=|f(x)-a| |g(x)-b|
Use triangle ineq, work it all out and you'll get that: For any [epsilon], when |x-c|<[delta] for suitable [delta],
|fg(x)-ab|< [epsilon]2 + |a|[epsilon] + |b|[epsilon]
After a little more work, the conclusion does hold.
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| « Last Edit: Mar 10th, 2004, 12:21pm by kellys » |
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