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March 10th, 2012, 10:01 PM   #1
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a limit to infinity

lim x(x-2)(x-4)(x-6)...(x-2k)/(x-1)(x-3)(x-5)...(x-(2k+1))
x-> infinity
How does the calculation of this depend on the choice of k (k is a positive integer)?
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March 10th, 2012, 10:20 PM   #2
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Re: a limit to infinity

The limit does not depend upon k. Think about what you learned in algebra/precalc about the horizontal asymptotes of rational functions. Feel free to ask for more information if this doesn't make it suddenly clear.
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March 11th, 2012, 05:01 PM   #3
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Re: a limit to infinity

Would the (x-2k) in the numerator be approaching infinity faster than the (x-(2k + 1)) in the denominator? If so, then does this whole equation mean that no matter what number I plug into k it will always end up at the same limit?

Last edited by skipjack; June 23rd, 2015 at 03:14 PM.
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March 11th, 2012, 06:36 PM   #4
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Re: a limit to infinity

What is the degree of the polynomial in the numerator in relation to the degree of the polynomial in the denominator?
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March 11th, 2012, 08:09 PM   #5
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Re: a limit to infinity

The degrees are the same. So this limit is just going to be infinity over infinity no matter what value k is?
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March 11th, 2012, 08:14 PM   #6
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Re: a limit to infinity

Good, yes the degrees are the same. Do you remember from your earlier courses how to find the horizontal asymptote of a rational function where the polynomial in the numerator has the same degree as the polynomial in the numerator?

When working with limits, the form ?/? is an indeterminate form. So to find the actual limit, we have several techniques we can use, but in this case we may think in terms of the horizontal asymptote. When the degrees are the same, we then simply take the ratio of the leading coefficients. What is this ratio?
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March 11th, 2012, 08:17 PM   #7
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Re: a limit to infinity

I've not done math for a long time so I'm not sure what horizontal asyptotes are. The ratio of the LCs would be 1/1 right?
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March 11th, 2012, 08:27 PM   #8
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Re: a limit to infinity

Yes, so the limit is 1. Here is an article on asymptotes:

http://en.wikipedia.org/wiki/Asymptote
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March 11th, 2012, 08:29 PM   #9
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Re: a limit to infinity

Cool, thanks man, you're a huge help.

Last edited by skipjack; June 23rd, 2015 at 03:16 PM.
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