Big-$O$ and Little-$o$ Notations¶

Definition¶

Consider two functions $f\colon \mathbb{R}\to\mathbb{R}$ and $g\colon \mathbb{R}\to\mathbb{R}\text{.}$ Let $a\in[-\infty,+\infty]$.

• We say that $f(x)$ is big-$O$ of $g(x)$ as $x$ tends to $a\text{,}$ and write $f(x)=O(g(x))$ as $x\to a\text{,}$ if there is a constant $M>0$ such that $|f(x)|\le M\,|g(x)|$ for all $x$ near $a\text{:}$ for $0<|x-a|<\delta$ (a punctured neighbourhood of $a$) when $a$ is finite, for $x>N$ when $a=+\infty\text{,}$ and for $x<-N$ when $a=-\infty\text{,}$ for some $\delta>0$ or $N\text{.}$
• Assume that $g(x)\neq0$ for all $x$ near $a$ in the same sense. We say that $f(x)$ is little-$o$ of $g(x)$ as $x$ tends to $a\text{,}$ and write $f(x)=o(g(x))$ as $x\to a\text{,}$ if $\displaystyle \lim\limits_{x\to a}\frac{f(x)}{g(x)}=0\text{.}$

The constant $M$ gives an Upper Bound on the relative size of $f$ near $a$. For little-$o$, use a Limit at a Point when $a$ is finite and a Limit at Infinity when $a=\pm\infty$ to test whether $f/g$ tends to zero.

Interpretation¶

Big-$O$ and little-$o$ compare the sizes of two functions near a point $a$ while ignoring constant factors. $f(x)=O(g(x))$ means that near $a$ the size of $f$ is at most a constant multiple of the size of $g$; $f(x)=o(g(x))$ means that $f$ is negligible compared with $g$ near $a$. When $g\to0$$0$, this reads: $f$ goes to zero at least as fast as $g$ (big-$O$), or faster than $g$ (little-$o$).

The constant $M$ in big-$O$ is fixed but may be large, so big-$O$ only records an order of magnitude: $3x^2=O(x^2)$ and $x^2=O(3x^2)$ as $x\to0$$0$. Little-$o$ allows no fixed constant: $f/g\to0$ means that for every $\varepsilon>0$ the bound $|f(x)|\le\varepsilon\,|g(x)|$ holds for all $x$ near $a$. Both notations depend on where $x$ goes: $x^2=o(x)$ as $x\to0$$0$, but $x=o(x^2)$ as $x\to+\infty$$+\infty$.

The main use is the remainder in Taylor's theorem: the error of the $k$-th order Taylor expansion is $o((x-a)^k)$ as $x\to a$$a$, small even compared with the last term kept. Comparing leading terms in this way also gives limits of quotients, alongside L'Hôpital's rule.

A common slip is to read the equals sign in $f(x)=O(g(x))$ as an equation. It is one-way shorthand for "$f$ is bounded by a constant multiple of $|g|$ near $a$". From $x^2=O(x)$ and $x^3=O(x)$ as $x\to0$ one cannot conclude $x^2=x^3$$x^3$, and the reversed form $O(g(x))=f(x)$ is not used.

The notation returns in Life Insurance, in the force of mortality, for the probability of dying within a very short time.

Key Results¶

• The little-$o$ notation makes a stronger assertion than the big-$O$ notation: every function that is little-$o$ of $g$ is also big-$O$ of $g\text{,}$ but not all functions that are big-$O$ of $g$ are little-$o$ of $g\text{.}$
• If $f(x)$ and $g(x)$ are asymptotically equivalent as $x$ tends to $a\text{,}$ i.e., $f(x)\sim g(x)$ as $x\to a\text{,}$ then we have $f(x)=O(g(x))$ and $g(x)=O(f(x))\text{,}$ as $x\to a\text{.}$

Examples¶

• As $x\to 0$$0$, $x^2$ is both $O(x)$ and $o(x)$ since $\displaystyle \lim\limits_{x \to 0}\frac{x^2}{x} = 0$$0$.
• As $x\to 0$$0$, $\sin (x^2)$ is $O(x^2)$ but not $o(x^2)$ since $\displaystyle \lim\limits_{x \to 0}\frac{\sin(x^2)}{x^2} = 1$$1$.

• In the following graph, we have $g(x)=o(f(x))$ and $g(x)=o(h(x))$ as $x\to 0$$0$:

  

  Near $0$ the parabola $g$ is much flatter than $f$ and $h$: $g(x)/f(x)\to0$ and $g(x)/h(x)\to0$ as $x\to0$$0$.

Prerequisites¶

• Upper Bound
• Limit at a Point
• Limit at Infinity

Related Concepts¶

• Asymptotic Equivalence
• Taylor's Theorem

Sources¶

• Math III: Analysis · Lecture 3

Aliases: Big-O, Big O, little-o, little o, Bachmann-Landau notation, Landau notation, Landau symbol Symbols: delta, element of, epsilon, infinity, real numbers

Last updated: 20 October 2026 03:02