# $f \in o(g), f \in \omega(g), f \sim g$ and existence of $\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)}$

1. Let f(x) and g(x) be two functions defined on some subset of the real numbers. There are two definitions for $f \in o(g)\mbox{ as }x\to\infty\,$, according to Wikipedia:

• $$\lim_{x \to \infty}\frac{f(x)}{g(x)}=0.$$
• for every $M > 0$, there exists a constant $x_0$, such that $$|f(x)| \le \; M |g(x)|\mbox{ for all }x>x_0.$$

I was wondering if the two definitions are equivalent? Can it be possible that the second one is more general than the first one in that the limit of the ratio may not exist?

2. Similar questions for $f \in \omega(g)$ and for $f \sim g$?

Thanks and regards!

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Isn't the second just spelling out what the limit means? – Fabian Mar 6 '11 at 16:19
@Fabian: I am not sure if the limit of the ratio always exist. For example, if no matter how close you approach $\infty$, can there always exist $x$ closer to infty so that $g(x)=0$? Similar question for f∈ω(g) and for f∼g? – Tim Mar 6 '11 at 16:22
Now I see what you mean. But what if you replace lim with limsup (wikipedia is not always the best place to get the definitions). – Fabian Mar 6 '11 at 16:25
In that spirit I would also say that there is an absolute value missing. What if we would rewrite the definition in Wikipedia (we have this power ;-)) and define little-o as $\limsup_{x\to\infty} \left| \frac{f(x)}{g(x)} \right| = 0$? – Fabian Mar 6 '11 at 16:28
@Fabian: (1) If replace lim with limsup, is taking absolute value of the ratio needed for the two definitions to be equivalent? (2) for f∈ω(g), do you also suggest to replace lim with liminf? how about for f∼g (3) Do you mind post the definition and its source that are the best you can find? – Tim Mar 6 '11 at 16:30

The two definitions of $“f(x)\in o(g(x)) \mbox{ as } x\to\infty”$ you stated are equivalent as long as $g(x)\neq0$ for every sufficiently large $x$. Usually we use the $O$-notation and its relatives only when this condition holds. Under this condition, the second definition is just what we mean by the first definition.
If this condition is not satisfied, then there are arbitrarily large points $x$ at which the function $f(x)/g(x)$ is not defined. In this case, it is not clear what we mean by $\lim_{x\to\infty}f(x)/g(x) = 0$.