I am interested in proving or finding a counter example for the following statement

$$ \lim_{n \to \infty}{\frac{\ln{f(n)}}{\ln{g(n)}}} = \infty \implies \lim_{n \to \infty}{\frac{f(n)}{g(n)}} = \infty $$

It seems to make a lot of sense, but it isn't very clear how to prove this statement.

If it isn't true, what if both $f(n)$ and $g(n)$ are monotonically increasing?

I've tried looking at the contrapositive, as well as Taylor series expansions, but I'm not able to come to a complete conclusion.

I would also imagine that this holds for any monotonic function, not just the logarithm (if it holds at all).

  • $\begingroup$ If $f(n)=2$ and $g(n)=1+\frac{1}{n}$, you have a counter example. $\endgroup$ Sep 12 '14 at 18:22

I already gave a counter example in the comments for the general case.

let's suppose that f and g are monotonically increasing and >1. Then their limits is $+\infty$.(otherwise, the limit of $\frac{ln(f(n))}{ln(g(n))}$ would not be $+\infty$)

Let's define h such as : $f(n)=h(n)g(n)$

Then : $\frac{ln(f(n))}{ln(g(n))}=\frac{ln(h(n)g(n))}{ln(g(n))}=1+\frac{ln(h(n)}{ln(g(n))}$. So you know that the limit of $\frac{ln(h(n)}{ln(g(n))}$ is $+\infty$.

Since the limit of $ln(g(n))$ is $+\infty$, the limit of $ln(h(n))$ is also $+\infty$.

Hence, by definition of $h$, the limit of $\frac{f(n)}{g(n)}$ is also $+\infty$.

Is it ok for you?

  • $\begingroup$ Very simple and beautiful answer. +1 $\endgroup$ Sep 13 '14 at 4:52
  • $\begingroup$ I think it is not necessary that $\log(g(n))$ also tends to $\infty$. It may tend to a finite limit. But your conclusion and proof holds in exactly the same manner. $\endgroup$ Sep 13 '14 at 4:56
  • $\begingroup$ Very good, thank you. $\endgroup$
    – RJTK
    Sep 13 '14 at 15:03

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