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Math Help - Polinomials and Limit

  1. #1
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    Polinomials and Limit

    Hello

    Could you PLEASE help me to solve the following exercise:

    Let p(x) and q(x) polinomials of order n and m respectively.

    Analyze what happens when p(x)/ q(x) and limx->0

    THANKS IN ADVANCE
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  2. #2
    Senior Member pankaj's Avatar
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    p(x)=a_{0}+a_{1}x+a_{2}x^2+.....+a_{n}x^n

    and

    q(x)=b_{0}+b_{1}x+b_{2}x^2+.....+b_{m}x^m


     <br />
\lim_{x\to 0}\frac{p(x)}{q(x)}=\frac{a_{0}}{b_{0}}<br />

    Nothing special about this.
    .................................................. ...............................................
    Now consider
    p(x)=a_{0}x^n+a_{1}x^{n+1}+a_{2}x^{n+2}2+.....+a_{  n}x^{n+k}

    q(x)=b_{0}x^m+b_{1}x^{m+1}+b_{2}x^{m+2}+.....+b_{n  }x^{m+l}

    Evaluate \lim_{x\to 0}\frac{p(x)}{q(x)} under the following conditions:

    (i) n=m

    (ii) n>m,n-m is even

    (iii) n<m,n-m is even, \frac{a_{0}}{b_{0}}>0

    (iv) n<m,n-m is even,  \frac{a_{0}}{b_{0}}<0
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  3. #3
    MHF Contributor
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    Quote Originally Posted by pankaj View Post
    p(x)=a_{0}+a_{1}x+a_{2}x^2+.....+a_{n}x^n

    and

    q(x)=b_{0}+b_{1}x+b_{2}x^2+.....+b_{m}x^m


     <br />
\lim_{x\to 0}\frac{p(x)}{q(x)}=\frac{a_{0}}{b_{0}}<br />

    Nothing special about this.
    .................................................. ...............................................
    Now consider
    p(x)=a_{0}x^n+a_{1}x^{n+1}+a_{2}x^{n+2}2+.....+a_{  n}x^{n+k}

    q(x)=b_{0}x^m+b_{1}x^{m+1}+b_{2}x^{m+2}+.....+b_{n  }x^{m+l}

    Evaluate \lim_{x\to 0}\frac{p(x)}{q(x)} under the following conditions:

    (i) n=m

    (ii) n>m,n-m is even

    (iii) n<m,n-m is even, \frac{a_{0}}{b_{0}}>0

    (iv) n<m,n-m is even,  \frac{a_{0}}{b_{0}}<0
    Why distinguish on whether n-m is even and whether

     <br />
\frac{a_0}{b_0} > 0\; \text{or}\; < 0?
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  4. #4
    Senior Member pankaj's Avatar
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    Just think over it
    Hint:what happens when you approach 0 from values less than 0.
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  5. #5
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    Whats the final answer?. Im confused
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  6. #6
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    Quote Originally Posted by pankaj View Post
    Just think over it
    Hint:what happens when you approach 0 from values less than 0.
    Right, the possibility of approaching the vertical asymptotes \pm \infty. My bad.
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  7. #7
    Senior Member pankaj's Avatar
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    Quote Originally Posted by osodud View Post
    Whats the final answer?. Im confused
    Give some time to it
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