# Limits of functions and convergent sequences.

Theorem 4.3.1   Let 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).

Noah Dana-Picard 2007-12-28