# asymptotically equal

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 limit holds:

This is often denoted f(*x*)~ g(*x*).

For example: 3*x*^{4}+2*x*+7 ~ 3*x*^{4}, *x*+sin *x* ~ *x*,
and the prime number theorem states that
pi(*x*) ~ *x*/ln *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)).

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