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Math Help - limits and conjectures

  1. #1
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    limits and conjectures

    Let P and Q be polynomials and consider the limit of P(x)/Q(x) as x approaches infinity. State and prove a conjecture for the value of this limit that depends on the degrees of P and Q and their leading coefficients.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by friday616 View Post
    Let P and Q be polynomials and consider the limit of P(x)/Q(x) as x approaches infinity. State and prove a conjecture for the value of this limit that depends on the degrees of P and Q and their leading coefficients.
    Let P=p_n x^n+p_{n-1}x^{n-1}+...+p_1 x+p_0 and let Q=q_m x^m+q_{m-1}x^{m-1}+...+q_1 x+q_0.

    If deg(P)>deg(Q), then \lim_{x\to\infty}\frac{P(x)}{Q(x)}=\infty.

    If deg(P)<deg(Q), then \lim_{x\to\infty}\frac{P(x)}{Q(x)}=0.

    If deg(P)=deg(Q), then \lim_{x\to\infty}\frac{P(x)}{Q(x)}=\frac{p_n}{q_m}.

    The proofs are fairly straightforward. If k=\min\{n,m\}, just factor out x^k from both polynomials and see what you have left.
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