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        riddles >> easy >> Limit of Product of Functions
        
(Message started by: william wu on Mar 10^th, 2004, 10:29am)

Title: Limit of Product of Functions
Post by william wu on Mar 10^th, 2004, 10:29am

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.

Title: Re: Limit of Product of Functions
Post by kellys on Mar 10^th, 2004, 12:15pm

[e]Assuming a,b,c are not infinity,[/e]
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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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