11.2 Limits

Recall that in Section 10.2 about sequences, we defined limits of convergent sequences. Intuitively, limits capture the notion that eventually, all elements get as close to the limit as we wish. This concept can be extended to functions as well.

Definition 51. (Limits of functions)

Let f : ℝ →ℝ be an arbitrary function. We say that

if for every sequence x_n →x₀, where x_n does not equal x₀ for all n,

holds.

Right off the bat, there are two essential things to note.

1. The limit of a function is defined in terms of limits of sequences. If all possible sequences of the form {f(x_n)} with x_n → x₀ have the same limit, then lim_x→x0f(x) is defined as the common limit.
2. With ∞-divergent sequences, limits at ±∞ are defined.

Here is a figure that helps visualize the process.

Figure 11.3: Illustration of limits: as x_n gets closer to 0, f(x_n) gets closer to lim_n→∞f(x_n)

To further illustrate...