1Differentiation
IA Differential Equations
1.2 Small o and big O notations
Definition (O and o notations).
(i)
“
f
(
x
) =
o
(
g
(
x
)) as
x → x
0
” if
lim
x→x
0
f(x)
g(x)
= 0. Intuitively,
f
(
x
) is much
smaller than g(x).
(ii)
“
f
(
x
) =
O
(
g
(
x
)) as
x → x
0
” if
f(x)
g(x)
is bounded as
x → x
0
. Intuitively,
f(x) is about as big as g(x).
Note that for f (x) = O(g(x)) to be true, lim
x→x
0
f(x)
g(x)
need not exist.
Usually,
x
0
is either 0 or infinity. Clearly, we have
f
(
x
) =
o
(
g
(
x
)) implies
f(x) = O(g(x)).
Note that this is an abuse of notation. We are not really saying that
f
(
x
)
is “equal” to
o
(
g
(
x
)), since
o
(
g
(
x
)) itself is not a function. Instead,
o
(
g
(
x
))
represents a class of functions (namely functions that satisfy the property above),
and we are saying that
f
is in this class. Technically, a better notation might
be
f
(
x
)
∈ o
(
g
(
x
)), but in practice, writing
f
(
x
) =
o
(
g
(
x
)) is more common and
more convenient.
Example.
– x = o(
√
x) as x → 0 and
√
x = o(x) as x → ∞.
– sin 2x = O(x) as x → 0 as sin θ ≈ θ for small θ.
– sin 2x = O(1) as x → ∞ even though the limit does not exist.
This notation will frequently be used in calculus. For example, if we want to
ignore all terms second order in
x
in an expression, we can write out the first
order terms and then append +
O
(
x
2
). In particular, we can use it to characterize
derivatives in a different way.
Proposition.
f(x
0
+ h) = f(x
0
) + f
0
(x
0
)h + o(h)
Proof. We have
f
0
(x
0
) =
f(x
0
+ h) − f(x
0
)
h
+
o(h)
h
by the definition of the derivative and the small
o
notation. The result follows.