Notation: Just curious

[Ishuda]

My math classes and degree were some time ago [more than half a century] so I'm somewhat out of date as far as 'standard notations' go with some things. Just out of curiosity, consider the notations o(x) and O(x) when speaking about the order of behavior of a function. Although they both mean (about ?) the same thing, that is the function behaves like (a constant times) x, is there any implication of size involved? BTW: What I mean by f(x) = o(g(x)) [or O(g(x))] is
\(\displaystyle \frac{f(x)}{g(x)}\) ~ constant
for x in some restricted neighborhood, i.e. close to zero or very large or ...

For example, take the polynominal
P(x) = a_n xⁿ + a_n-1 x^n-1 + ... + a₁ x + a₀
where a_n is not zero and a₀ is zero. When I 'was growing up', 'short cut notation' was P = o(x) meant that P(x) behaved like x when x 'was small' and P = O(xⁿ) meant that P behaved like xⁿ when x 'was large'. [Unless, of course, something else was specified about x.] Is that still the case?

 

[HallsofIvy]

I think you must be mis-remembering. I'm no youngster myself but I don't believe those were ever the meanings of \(\displaystyle f= O(g)\) and \(\displaystyle f= o(g)\). Rather, you must specify "as x goes to a" with a= 0 or \(\displaystyle \pm \infty\) or even some finite non-zero number. \(\displaystyle f= O(g)\) means \(\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)}\) is a non-zero constant. \(\displaystyle f= o(g)\) means \(\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)}= 0\).

 

[Ishuda]

...Rather, you must specify "as x goes to a"

Yes there is that. and that is actually what i meant to imply, i.e it is as close as you want to get the constant if the neighborhood is small enough [slightly different meaning if speaking of unbounded neighborhoods of course].

As far as the other part goes. maybe I am misremembering so I'll just need to say what I mean in those cases instead of using the notation.