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Math Help - big-o, but long division first

  1. #1
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    big-o, but long division first

    How is long division done on something like this. The log is what I don't understand:


    This problem is actually for a discrete math course. The problem says we must find the least n such that f(x) is O(x^n), but for now I'd just like to understand the long division.

    Though, an answer and explanation for n, and witnesses C and k would be cool too.
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  2. #2
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    One does not need long division here. In any case, I have not seen long division applied to non-polynomials, though there may be a way to extend it.

    In rate of growth problems, log(x) is a little brother of x. Also, the rate of a polynomial is determined by its degree. Informally, the nominator here is dominated by x^4+5x, which is approximately x^4. The denominator is also approximately x^4, so the whole function is O(1).

    More formally, you can write a series of upper bounds, e.g.: (x^4+5\log x)/(x^4+1)\le(x^4+5\log x)/x^4\le(x^4+5x)/x^4\le\dots. Some of those inequalities may not hold for all x, but only for x > k for some k.
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  3. #3
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    How can you determine a constant C such that f(x) \leq C*g(x)?

    would I continue with the bounds until I reached  \frac {x^4 + x^4}{x^4} which would yield 2?
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  4. #4
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    Yes, this will work. Note that 5x\le x^4 is false for x = 1 but is true for x\ge 2.
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