Inverses of asymptotic functions

[Palafox]

Suppose f(x) and g(x) are monotone increasing functions (continuous, and smooth if necessary) which are asymptotic -- that is, their quotient has limit 1 as x→∞. Are their inverses asymptotic?

 

[mfb]

Hint: Look for counter-examples.

 

[Palafox]

Right, mfb! For example, log x and log 2x. What I should have asked was, under what circumstances...

 

[mfb]

That is nearly the example I found (log x and 1 + log x). I think you need those slowly/quickly growing functions, where a small difference in one variable (vanishes in the limit) can lead to a large difference in the other one.