I have seen the O and o notation used in calculating limits, as it is a great way to simplify the problem, as it provides some sort of mathematical approximation. %%visits: 2 O notation is also used to find out which functions grow faster than the others.
Think of o_notation as error terms, that eventually tend to zero so you do not need to worry about them. It is some function that goes to zero faster then it’s class.
Little o notation means that these terms behave like this simpler function as x gets really close to something (infinitely close)
Big O notation means that these terms can be controlled by a smaller function, and don’t blow up (stay bounded) as the function gets fairly close to (no limit here) x.
Big 0 converges the ratio of sequences to a constant, but little o means that constant is always 0, so little o implies big o but the converse is false
There is a calculus of o, but this sounds much more complicated that it needs to, just stick to the definitions and the weird bringing tn to the other side and that helps
f(x) = O(g(x)) as x → x0, if $\frac{f(x)}{g(x)}$ is bounded for small x − x0; aka C1f(n) ≤ g(n) ≤ c2f(n) f(x) = o(g(x)) as x → x0, if $\lim_{x \to x_0} \left[ \frac{f(x)}{g(x)} \right] = 0$
E.g. as x → 0 - xn = o(xm), n > m The smaller power takes over closer to 0, think 0.52 is bigger than 0.520 - O(xn) + O(xm) = O(xm) The smaller power is insignificant - O(xn)O(xm) = O(xn + m) Normal multiplication of powers - O(xn) = o(1) Any function of this form is going to tend to 0 as x → 0, limx → 0f(x) = 0 - log |x| = O(x−1)
E.g. as x → ∞ - xm = o(xn), n > m The larger power takes over - O(xn)O(xm) = O(xn + m) - O(x−n) = o(1), n > 0 - x = O(eax), a > 0
Asymptotic regime :~ a way of saying it needs to be tending to something
The O and o notation Let f and g be two functions defined on an open interval Δ ⊂ ℝ, and let x0 ∈ Δ. Suppose g(x) ≠ 0 for small x − x0 ≠ 0 We write
$f(x) = O(g(x)) \text{ as } x \to x_0 \text{ , if } \frac{f(x)}{g(x)} \text{ is bounded for small } x - x_0$ f(x) = o(g(x)) x x_0, if _{x x_0} = 0 $$
The same notation can be used if x → ∞ or x → −∞
o(1) just means it tends to 0 by itself # examples as x → 0 - O(xn)O(xm) = O(xn + m) - sin x = O(x) as |sin x| ≤ |x| - $\log |x| = O(\frac{1}{x})$ as x → ∞ - x = o(x2) as x → ∞ :todo: throw in some calculus limit questions to see it in action. tags :math:real_analysis: