This page is a sub-page of our page on Limits.

///////

Related KMR pages:

• Differentiation and Affine Approximation in One Real Variable
• Taylor expansion in One Real Variable

///////

Other relevant sources of information:

• Big-Ordo notation
• Little-ordo notation

///////

Big-Ordo and little-ordo

The symbols \, O \, = “big Ordo” and \, o \, = “little ordo” describe how fast functions are decreasing towards zero when their independent variable is approaching some fixed point where they take the value zero.

Let \, f : \mathbb{R} \rightarrow \mathbb{R} \, be a function
and let its independent variable \, x approach a fixed point \, a \in \mathbb{R} where \, f(a) = 0 .

Definition: \, O_a(f) \stackrel {\mathrm{def}}{=} \{ g : \mathbb{R} \rightarrow \mathbb{R} \} \, such that, whenever \, x \, is close enough to \, a ,
we have \, | \,\dfrac{g(x)}{f(x)} \, | < M \} \, for some fixed, positive constant \, M \in \mathbb{R} .

Definition: \, o_a(f) \stackrel {\mathrm{def}}{=} \{ g : \mathbb{R} \rightarrow \mathbb{R} \} \, such that, whenever \, x \, is close enough to \, a ,
we have \, | \,\dfrac{g(x)}{f(x)} \, | < M \} \, for every fixed, positive constant \, M \in \mathbb{R} .

Intuitively, this means that:

\, O_a(f) \, denotes the set of functions that decrease at least as fast as \, f \,
when \, x \, approaches \, a ,
while
\, o_a(f) \, denotes the set of functions that decrease faster than \, f \,
when \, x \, approaches \, a .

NOTATION: When the point that is approached is clear from the context,
we write \, O(f) \, and \, o(f) \, instead of \, O_a(f) \, and \, o_a(f) .

///////

Rules of computation for \, O(f) \, :

1) \, f \cdot O(g) = O(f \cdot g) \,

2) \, O(f) \cdot O(g) = O(f \cdot g) \,

3) \, O(O(f)) = O(f) \,

In words:

1) A function \, f \,
times any function that decreases at least as fast as the function \, g \,
must decrease at least as fast as the function \, f \cdot g .

2) Any function that decreases at least as fast as the function \, f \,
times any function that decreases at least as fast as the function \, g \,
must decrease at least as fast as the function \, f \cdot g .

3) Any function that decreases at least as fast as
any function that decreases at least as fast as the function \, f \,
must decrease at least as fast as the function \, f .

///////

Rules of computation for \, o(f) \, :

1) \, f \cdot o(g) = o(f \cdot g) \,

2) \, o(f) \cdot o(g) = o(f \cdot g) \,

3) \, o(o(f)) = 0 \,

In words:

1) A function \, f \,
times any function that decreases faster than the function \, g \,
must decrease faster than the function \, f \cdot g .

2) Any function that decreases faster than the function \, f \,
times any function that decreases faster than the function \, g \,
must decrease faster than the function \, f \cdot g .

3) Any function that decreases faster than
any function that decreases faster than the function \, f \,
must be identically zero in some neighborhood of the point that is approached.

///////

Example:

The interactive simulation that created this movie

The blue line intersects the red parabola tangentially at the point \, P .

The green line intersects the red parabola transversally at the point \, P .

The blue point \, W \, approaches \, P \, along the blue line,
which intersects the parabola tangentially at \, P .

The black point \, Q \, approaches \, P \, along the parabola itself.

The green point \, R \, approaches \, P \, along the green line,
which intersects the parabola transversally at \, P .

The horizontal distance between the green and the red vertical lines is \, ∆ \, .

TAKE AWAY MESSAGE:

\, | \, QW \, | = o(∆)
\, | \, RW \, | = O(∆)

In general:

Let the parabola be replaced by any smooth curve
and fix a point \, P \, on this curve. Then the following is true:

Any point \, Q \, that approaches \, P \, along a curve
that intersects the given curve tangentially at the point \, P \,
behaves like the point \, Q \, in the movie, i.e., we have \, | \, QW \, | = o(∆) .

Any point \, R \, that approaches \, P \, along a curve
that intersects the given curve transversally (= non-tangentially) at the point \, P \,
behaves like the point \, R \, in the movie, i.e., we have \, | \, RW \, | = O(∆) .