Limits of functions and convergent sequences.

Theorem 4.3.1   Let $ f$ be a function defined on a pointed neighborhood of $ x_0$ . The function $ f$ has a limit $ l$ when $ x$ is arbitrarily close to $ x_0$ if, and only if, for any convergent sequence $ (x_n)$ having $ x_0$ as its limit, we have:

$\displaystyle \underset{n \underset{>}{\rightarrow} + \infty}{\lim} f(x_n) = f(l).$    

Proof. $ \qedsymbol$

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

Noah Dana-Picard 2007-12-28