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based on the result of finding the limit

We consider three cases in turn.

(a) Little-oh
We write

if the limit, (1.31), is L = 0; we say that ‘ f is little-oh of g as Clearly, this
property of a function does not provide very useful information; essentially all it
says is that f (x) is smaller than g (x) (as So, for example, we have

but also

and
It is an elementary exercise to show that each satisfy the definition L = 0 from
(1.31), by using familiar ideas that are typically invoked in standard ‘limit’ problems.
For example, the last example above involves

confirming that the limit is zero. (Note that, in the above examples, the gauge func-
tion which is a non-zero constant is conventionally taken to be g (x) = 1; note also
that the limit under consideration should always be quoted, or at least understood.)
(b) Big-oh
We write

if the limit, (1.31), is finite and non-zero; this time we say that ‘ f is big-oh of g
as or simply ‘ f is order g as As examples, we offer

but

also