My Math Forum proof of limits to infinity of rational functions?
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 June 27th, 2010, 11:42 PM #1 Member   Joined: Aug 2009 Posts: 69 Thanks: 0 proof of limits to infinity of rational functions? basically, I've heard this all over the place: lim n-> $\infty$ of f(n)/g(n) where f and g are polynomial functions = ... 0 if the degree of f is less than the degree of g or positive or negative infinity if the degree of f is greater than the degree of g or the ratio of the coefficients of the terms with largest exponents in f and g but i've never seen a proof of that... and I really don't care to know something without knowing how to prove it, unless I am truly truly too far in over my head, which I'm not sure I am.
 June 28th, 2010, 04:59 AM #2 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: proof of limits to infinity of rational functions? If f and g are of the same degree, and x is not zero, then by dividing both top and bottom by x? you get: $\frac{f}{g}=\frac{ax^n+bx^{n-1}+cx^{n-2}+...}{\alpha x^n+\beta x^{n-1}+\gamma x^{n-2}+...}=\frac{a+\frac{b}{x}+\frac{c}{x^2}+...}{\al pha+\frac{\beta}{x}+\frac{\gamma}{x^2}+...}$ and as x approaches infinity, both top and bottom approach finite limits, so the limit of f/g is $a/\alpha$.
 June 28th, 2010, 05:59 PM #3 Member   Joined: Aug 2009 Posts: 69 Thanks: 0 Re: proof of limits to infinity of rational functions? Ah! I see! cool! and the proofs for the other two cases are simple enough. Thanks!

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# proving rational limits

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