Limits

Definition

We call x the limit of the sequence (x_n) if the following condition holds:

  • For each real number \epsilon > 0, there exists a natural number N such that, for every natural number n > N, we have |x_n - x| < \epsilon.

In other words, for every measure of closeness \epsilon, the sequence’s terms are eventually that close to the limit. The sequence (x_n) is said to converge to or tend to the limit x, written x_n \to x or \lim_{n \to \infty} x_n = x.

If a sequence converges to some limit, then it is convergent; otherwise it is divergent.

Example

  • If x_n = c for some constant c, then x_n \to c

Properties

Let \lim\limits_{x \rightarrow a} f(x) = L_1, \ \lim\limits_{x \rightarrow a} g(x) = L_2

  • \lim\limits_{x \rightarrow a} [f(x) \pm g(x)] = L_1 \pm L_2
  • \lim\limits_{x \rightarrow a} [f(x) \cdot g(x)] = L_1 \times L_2
  • \lim\limits_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{L_1}{L_2}, \ L_2 \neq 0
  • \lim\limits_{x \rightarrow a} f(x)^a = L_1^a

Notation

\lim_{n \to \infty} x_n = a