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Solving a Limit Problem

  1. Jan 31, 2015 #1
    Suppose there is a limit
    ##\lim_{n \to \infty} \frac{n^{1.74}}{n \times (\log n)^9}##
    Taking logs both on numerator and denominator
    ##=\lim_{n \to \infty} \frac{1.74 \times \log n}{\log n + 9 \log \log n}##
    What can we say about the limit as n approaches ##\infty##
  2. jcsd
  3. Jan 31, 2015 #2
    How can you take log on both numerator and denominator. You are changing the question.
    1/2= 0.5 is not the same as
    log(1)/log(2) , which is 0.
  4. Feb 1, 2015 #3
    In order to compare growth rate of two functions we can do that.
  5. Feb 1, 2015 #4
    Can you give an example?
    Algebraically , in question you have stated how one can do that on numbers?
    Is that a university( college ) level problem
    Or a high school one?
  6. Feb 1, 2015 #5


    Staff: Mentor

    In your first post (quoted above) you are claiming that ##\lim_{n \to \infty} \frac{n^{1.74}}{n \times (\log n)^9} = \lim_{n \to \infty} \frac{1.74 \times \log n}{\log n + 9 \log \log n}##. Most emphatically, this is NOT TRUE! You can take the log of both sides of an equation, but you cannot take the log of a quotient to get a quotient of logs, nor can you take the log of a sum and get the sum of the logs.

    What IS true is that log(a * b) = log(a) + log(b), and that log(a/b) = log(a) - log(b), assuming suitable restrictions on the values of a and b.
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