Theorem 4.3.1Let
be a function defined on a pointed neighborhood of
. The function
has a limit
when
is arbitrarily close to
if, and only if, for any convergent sequence
having
as its limit, we have:

Proof.

Theorem 3.1 is mainly use to disprove the existence of a limit at some point, as shown in the next example.

Example 4.3.2
Let
. This function has no limit at 0:

Take
. Then
and
.

Take
. Then
and
.

As the limit of a function is unique ( v.s. Prop. 1.2 ),
it cannot be equal both to 0 and to 1, therefore
has no limit at 0 (in Figure 8 we show two drawings of the graph of
, zooming to show how ``messy'' looks the graph for
close to 0).