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asymptotically equal (another Prime Pages' Glossary entries) |
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Glossary:
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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: 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)).
Chris Caldwell © 1999-2009 (all rights reserved)
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