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How common it is to write e.g. $1-o(1)$ for a function that eventually approaches $1$ from below (or eventually equals $1$)? Would a better notation be $1-|o(1)|$ or what is meant is already obvious from $1-o(1)$ ? Obviously, precisely defining everything will work eventually, but I was wondering about the general convention. Similarly, are notations like $|\omega(1)|$ common?

Edit: In other words, would you say "What the hell is this?" is you saw both $1+o(1)$ and $1-o(1)$ in the same sentence, or would you say "Hmm, ok I got it, these are all assumed to be positive"?

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up vote 1 down vote accepted

The definition of little-Oh I recall is: $$ f=o(g) \Leftrightarrow \exists\varepsilon, \varepsilon\to 0, f=\varepsilon g $$ In particular, it doesn't really make sense to write "$-o(1)$"; according to this definition, if $f$ is $o(g)$, $(-f)$ is also $o(g)$. Another convention to circumvent that is to write $f\to 1^-$ or $f\to 1^+$ (if the limit is reached resp. from below or above). Or $\lim f= 1^-$ ($\lim f= 1^+$).

Rk: I've also seen teachers write $f\nearrow 1$ and $f\searrow 1$, but I'm not sure it is standard.

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Belated thanks for the answer. Let me also note that I have seen some papers that say $1-o(1)$ (especially in probability). I also used the notation $1-o(1)$ in my paper myself, nobody has said anything yet :) – Lord Soth Jul 2 '13 at 4:01

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