Little-o Notation — Definition, Formula & Examples

Little-o notation describes a function that grows strictly slower than another function. Writing

f(x) = o(g(x))

means

f(x)

becomes negligible compared to

g(x)

x

approaches some limit.

We write $f(x) = o(g(x))$ as $x \to a$ if $\displaystyle\lim_{x \to a} \frac{f(x)}{g(x)} = 0$ . This means that for every $\epsilon > 0$ , there exists a neighborhood of $a$ in which $|f(x)| \leq \epsilon\,|g(x)|$ .

Key Formula

f(x) = o(g(x)) \;\text{ as } x \to a \iff \lim_{x \to a} \frac{f(x)}{g(x)} = 0

Where:

$f(x)$ = The function claimed to be asymptotically smaller
$g(x)$ = The comparison function
$a$ = The point or direction of the limit (could be 0, ∞, etc.)

How It Works

To check whether

f(x) = o(g(x))

, compute the limit of

\frac{f(x)}{g(x)}

. If the limit equals zero, the little-o relationship holds. The point

a

is often

0

\infty

, or another value relevant to the problem. Little-o is stronger than Big-O:

f = o(g)

implies

f = O(g)

, but not vice versa. In Taylor expansions, little-o captures the idea that the remainder term shrinks faster than the last included power of

(x - a)

Worked Example

Problem: Show that

x^2 = o(x)

x \to 0

Form the ratio: Divide

f(x) = x^2

g(x) = x

\frac{x^2}{x} = x

Take the limit: Evaluate the limit as

x \to 0

\lim_{x \to 0} x = 0

Conclude: Because the limit is

0

, the definition is satisfied.

x^2 = o(x) \text{ as } x \to 0

Answer:

x^2 = o(x)

x \to 0

, confirming that

x^2

shrinks much faster than

x

near zero.

Why It Matters

Little-o notation appears throughout calculus and analysis whenever you need to describe how fast a remainder or error term vanishes. It is the standard way to write Taylor's theorem with the Peano form of the remainder, e.g.,

e^x = 1 + x + \frac{x^2}{2} + o(x^2)

. In algorithm analysis, it also distinguishes functions that are strictly slower-growing from those that merely share an upper bound.

Common Mistakes

Mistake: Confusing little-o with Big-O. Students sometimes treat

o(g)

and

O(g)

as interchangeable.

Correction: Big-O (

O

) means the ratio

f/g

stays bounded; little-o (

o

) means the ratio goes to zero. Little-o is the strictly stronger condition.