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12 1. Mathematical preliminaries

finally

but

(Little-oh and big-oh–o and O—are usually called the Landau symbols.)
(c) Asymptotically equal to or behaves like
Finally, we write

if the limit L, in (1.31), is precisely L = 1; then we say that ‘ f is asymptotically
equal to g as or ‘ f behaves like g as Some examples are

and then we may also write

Finally, it is not unusual to use ‘=’ in place of ‘~’, but in conjunction with a
measure of the error. So, with ‘~’, ‘O’ and ‘o’ as defined above, we write

or

but such statements should be regarded as no more than equivalents to some of
the statements given earlier. Some exercises that use o, O and ~ are given in
Q1.6, 1.7 and 1.8.
We should comment that other definitions exist for O, for example, although
what we have presented is, we believe, the most straightforward and most directly
useful. An alternative, in particular, is to define f (x) = O[g(x)] as if
positive constants C and R s.t.

our limit definition follows directly from this.