Degree of a Rational FunctionDate: 06/19/2003 at 08:28:35 From: Ban Nguyen Subject: Rational function f(x) = p(x)/q(x) You know that if the highest degree of p(x) is smaller than the highest degree of q(x), we have that the horizontal is y=0. Can you prove that? Date: 06/19/2003 at 15:00:48 From: Doctor Rob Subject: Re: Rational function Thanks for writing to Ask Dr. Math. Suppose that the degree of p is n and that of q is m. Then write n n p(x) = SUM p[i]*x^(n-i) = x^n*SUM p[i]/x^i, i=0 i=0 m m q(x) = SUM q[i]*x^(m-i) = x^m*SUM q[i]/x^i. i=0 i=0 Then n m lim f(x) = lim x^(n-m)* lim {SUM p[i]/x^i}/{SUM q[i]/x^i}. x->oo x->oo x->oo i=0 i=0 Now the last limit is easily seen to be p[n]/q[m], a finite nonzero constant. Since n < m, the preceding limit is easily seen to be 0. Feel free to write again if I can help further. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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