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Polynomial Limits

  1. Jan 5, 2006 #1


    My Calculus professor has indicated a 'shortcut' in determining polynomial fraction limits, I am inquiring if this identity is correct, and how comprehensive is this 'theory'?

    Polynomial Limit Theorem:
    [tex]\lim_{x \rightarrow \infty} \frac{ax^2 - x + 2}{bx^2 - 1} = \frac{a}{b}[/tex]

     
    Last edited: Jan 5, 2006
  2. jcsd
  3. Jan 5, 2006 #2

    Tx

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    Just divide through by the highest power, then the limit becomes A/B as x -> oo.
     
    Last edited by a moderator: Jan 5, 2006
  4. Jan 6, 2006 #3

    VietDao29

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    Homework Helper

    You can also expand your Polynomial Limit Theorem like this:
    Let [itex]m , \ n \in \mathbb{Z ^ +}[/itex]
    If m < n:
    [tex]\lim_{x \rightarrow \infty} \frac{\sum \limits_{i = 0} ^ m a_i x ^ i}{\sum \limits_{k = 0} ^ n b_k x ^ k} = 0 \quad (a_m, b_n \neq 0)[/tex]
    If m > n:
    [tex]\lim_{x \rightarrow \infty} \frac{\sum \limits_{i = 0} ^ m a_i x ^ i}{\sum \limits_{k = 0} ^ n b_k x ^ k} \quad (a_m, b_n \neq 0)[/tex] it does not have a limit.
    If m = n:
    [tex]\lim_{x \rightarrow \infty} \frac{\sum \limits_{i = 0} ^ m a_i x ^ i}{\sum \limits_{k = 0} ^ n b_k x ^ k} = \frac{a_m}{b_n} \quad (a_m, b_n \neq 0)[/tex]
     
    Last edited: Jan 6, 2006
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