One way of saying that two functions f(x) and
g(x) are about the same size is to say that they
are asymptotically equal: Two functions
f(x) and g(x)are asymptotically equal
(as x approaches infinity) if the following
This is often denoted f(x)~ g(x).
For example: 3x4+2x+7 ~ 3x4, x+sin x ~ x,
and the prime number theorem states that
pi(x) ~ x/log x.
If f(x)~g(x), then f is O(g) and
g is O(f),
but the converse is not true. Equivalently, if
f(x)~ g(x), then g has the same order
of magnitude as f (and again the converse is false).
Finally, if f(x)~ g(x), then
f(x)- g(x) is o(g(x)).
See Also: BigOh, LittleOh