MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Say we have two elementary functions (see for the definition) $f_1,f_2\colon [0,\infty)\mapsto \mathbb{R}$ such that $\lim\limits_{x\to\infty}f_1(x)=\lim\limits_{x\to\infty}f_2(x)=\infty$. Can we say something about the existence of $\lim\limits_{x\to\infty}\frac{f_1(x)}{f_2(x)}$? I heard that some kind of answer gives theorem proved in 1930's by GH Hardy but I couldn't find it.

share|cite|improve this question
That theorem relates to the "Hardy field" (See, not to elementary functions in the sense of Liouville. The latter have no such property: just take $f_1=x(2+\sin x)$ and $f_2=x$. – fedja Nov 2 '12 at 10:49
Yes. Note that even the muddled definition on the mathworld web site says that trigonometric functions are included among "elementary functions". – Gerald Edgar Nov 2 '12 at 15:13

In the book by Hardy Orders of Infinity you will find the Theorem (p. 18)

Any L-function is ultimately continuous, of constant sign, and monotonic, and, as $x\to\infty$, tends to $\infty$, or to zero or to some other definite limit. Further, if $f$ and $\phi$ are L-functions, one or other of the relations $f\succ\phi$, $f\asymp\phi$, $f\prec\phi$ holds between them.

With the definition:

We define a logarithmic-exponential function (shortly, an L-function) as a real one valued function defined, for all values of $x$ greater than some definite value, by a finite combination of the ordinary algebraical symbols (viz. $+$, $-$, $\times$, $\div$, $\root n \of \cdot$ ) and the functional symbols $\log(\cdots)$ and $e^{(\dots)}$, operating on the variable $x$ and on real constants.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.