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Math Help - Big O and Growth of Functions

  1. #1
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    Big O and Growth of Functions

    Hey I can't figure this HW assignment out, could anyone help me solve this...it would greatly be appreciated.

    Find the least integer n such that f(x) is O(x^n) for each of these functions.

    a) f(x) = 2x^2+x^3logx
    b) f(x) = 3x^5+(logx)^4
    c) f(x) = (x^4+x^2+1)/(x^4+1)
    d) f(x) = (x^3+5logx)/(x^4+1)
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Dharper07 View Post
    Hey I can't figure this HW assignment out, could anyone help me solve this...it would greatly be appreciated.

    Find the least integer n such that f(x) is O(x^n) for each of these functions.

    a) f(x) = 2x^2+x^3logx
    b) f(x) = 3x^5+(logx)^4
    c) f(x) = (x^4+x^2+1)/(x^4+1)
    d) f(x) = (x^3+5logx)/(x^4+1)
    First do you have a definition of what:

    f(x)=O(x^n)

    means?

    It means that there exists a positive constant k there exists an x_0 such that for all x>x_0

    |f(x)|<k |x^n|

    a) f(x) = 2x^2+x^3\log(x)

    Here clearly there is no k such that eventualy:

     <br />
|2x^2+x^3\log(x)|<k|x^3|<br />

    since \log(x) also grows without bound. But as \log(x) growns more slowly than x (that is \lim_{x \to \infty} \log(x)/x =0) we can find k such that:

     <br />
|2x^2+x^3\log(x)|<k|x^4|<br />

    So the least integer n such that:

    2x^2+x^3\log(x)=O(x^n)

    is 4

    CB
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