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Limit of Ratio of Two Functions


Mar 5, 2012
Hello everyone!

I'm considering the problem of comparing the asymptotic behavior of two functions $f(x)$ and $g(x)$.

We can say that if $\lim _{x\rightarrow \infty} f(x) / g(x) = \infty$ then f is asymptotically greater than g. Similarly, if this limit is 0, then f is asymptotically less than g.

Now, how about if use $\lim _{x\rightarrow \infty} \log (f(x) / g(x))$, if the limit is $\infty$, then f is greater, and if the limity is $-\infty$, f is smaller, right?

Can we conclude from $\lim _{x\rightarrow \infty} \log f(x) / \log g(x)$ something about the respective behavior of f and g? It sure does help to look at logs of fast-growing functions...


Well-known member
MHB Math Helper
Jan 27, 2012
Re: Limit of Ration of Two Functions

The logarithm is monotonic increasing and changes no relationship. This is why logarithms (and square roots) are popular transformations.