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Math Help - help with finding limit

  1. #1
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    help with finding limit

    Could someone show me how to do these limits? Let p(x) be a polynomial of degree n: p(x) = a0 + a1x + a2x^2 + + anx^n (an cannot equal 0). How would you prove that the limit from x to infinity of p(x)/anx^n is 1 and that the limit from x to negative infinity of p(x)/anx^n is 1.

    Thank you for any help.
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  2. #2
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    Quote Originally Posted by MKLyon View Post
    Could someone show me how to do these limits? Let p(x) be a polynomial of degree n: p(x) = a0 + a1x + a2x^2 + + anx^n (an cannot equal 0). How would you prove that the limit from x to infinity of p(x)/anx^n is 1 and that the limit from x to negative infinity of p(x)/anx^n is 1.

    Thank you for any help.
    For the first one:
    \lim_{x \to \infty} \frac{p(x)}{a_n x^n}

    = \lim_{x \to \infty} \frac{a_n x^n + a_{n - 1}x^{n - 1} + ~ ... ~ + a_1x + a_0}{a_n x^n}

    Divide the numerator and denominator by x^n:
    = \lim_{x \to \infty} \frac{a_n + a_{n - 1}x^{-1} + ~ ... ~ + a_1x^{-n + 1} + a_0x^{-n}}{a_n}

    = \lim_{x \to \infty} \left ( \frac{a_n}{a_n} + \frac{a_{n - 1}}{a_n} \frac{1}{x} + ~ ... ~ + \frac{a_1}{a_n} \frac{1}{x^{n - 1}} + \frac{a_0}{a_n} \frac{1}{x^n} \right )

    = 1

    You do the second one.

    -Dan
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