2.10: Appendix - Limits

The intuitive idea behind limits is relatively simple. Still, in the 19th century mathematicians were troubled by the lack of rigor, so they set about putting limits and analysis on a firm footing with careful definitions and proofs. In this appendix we give you the formal definition and connect it to the intuitive idea. In 18.04 we will not need this level of formality. Still, it’s nice to know the foundations are solid, and some students may find this interesting.

Limits of Sequences

Intuitively, we say a sequence of complex numbers \(z_1, z_2, ...\) converges to \(a\) if for large \(n\), \(z_n\) is really close to \(a\). To be a little more precise, if we put a small circle of radius \(\epsilon\) around \(a\) then eventually the sequence should stay inside the circle. Let’s refer to this as the sequence being captured by the circle. This has to be true for any circle no matter how small, though it may take longer for the sequence to be ‘captured’ by a smaller circle.

This is illustrated in Figure \(\PageIndex{1}\). The sequence is strung along the curve shown heading towards \(a\). The bigger circle of radius \(\epsilon_2\) captures the sequence by the time \(n = 47\), the smaller circle doesn’t capture it till \(n = 59\). Note that \(z_{25}\) is inside the larger circle, but since later points are outside the circle we don’t say the sequence is captured at \(n = 25\).

Figure \(\PageIndex{1}\): A sequence of points converging to \(a\). (CC BY-NC; Ümit Kaya)

Definition

The sequence \(z_1, z_2, z_3, ...\) converges to the value \(a\) if for every \(\epsilon > 0\) there is a number \(N_{\epsilon}\) such that \(|z_n - a| < \epsilon\) for all \(n > N_{\epsilon}\). We write this as

\[\lim_{n \to \infty} z_n = a. \nonumber \]

Again, the definition just says that eventually the sequence is within \(\epsilon\) of \(a\), no matter how small you choose \(\epsilon\).

Example \(\PageIndex{1}\)

Show that the sequence \(z_n = (1/n + i)^2\) has limit -1.

Solution

This is clear because \(1/n \to 0\). For practice, let’s phrase it in terms of epsilons: given \(\epsilon > 0\) we have to choose \(N_{\epsilon}\) such that

\[|z_n - (-1)| < \epsilon \text{ for all } n > N_{\epsilon} \nonumber \]

One strategy is to look at \(|z_n + 1|\) and see that \(N_{\epsilon}\) should be. We have

\[|z_n - (-1)| = \left|(\dfrac{1}{n} + i)^2 + 1\right| = \left|\dfrac{1}{n^2} + \dfrac{2i}{n}\right| < \dfrac{1}{n^2} + \dfrac{2}{n} \nonumber \]

So all we have to do is pick \(N_{\epsilon}\) large enough that

\[\dfrac{1}{N_{\epsilon}^2} + \dfrac{2}{N_{\epsilon}} < \epsilon \nonumber \]

Since this can clearly be done we have proved that \(z_n \to i\).

This was clearly more work than we want to do for every limit. Fortunately, most of the time we can apply general rules to determine a limit without resorting to epsilons!

Remarks

1. In 18.04 we will be able to spot the limit of most concrete examples of sequences. The formal definition is needed when dealing abstractly with sequences.
2. To mathematicians \(\epsilon\) is one of the go-to symbols for a small number. The prominent and rather eccentric mathematician Paul Erdos used to refer to children as epsilons, as in ‘How are the epsilons doing?’
3. The term ‘captured by the circle’ is not in common usage, but it does capture what is happening.

\(\lim_{z \to z_0} f(z)\)

Sometimes we need limits of the form \(\lim_{z \to z_0} f(z) = a\). Again, the intuitive meaning is clear: as \(z\) gets close to \(z_0\) we should see \(f(z)\) get close to \(a\). Here is the technical definition

Definition

Suppose \(f(z)\) is defined on a punctured disk \(0 < |z - z_0| < r\) around \(z_0\). We say \(\lim_{z \to z_0} f(z) = a\) if for every \(\epsilon > 0\) there is a \(\delta\) such that

\(|f(z) - a| < \epsilon\) whenever \(0 < |z - z_0| < \delta\)

This says exactly that as \(z\) gets closer (within \(\delta\)) to \(z_0\) we have \(f(z)\) is close (with \(\epsilon\)) to \(a\). Since \(\epsilon\) can be made as small as we want, \(f(z)\) must go to \(a\).

Remarks

1. Using the punctured disk (also called a deleted neighborhood) means that \(f(z)\) does not have to be defined at \(z_0\) and, if it is then \(f(z_0)\) does not necessarily equal \(a\). If \(f(z_0) = a\) then we say the \(f\) is continuous at \(z_0\).
2. Ask any mathematician to complete the phrase “For every \(\epsilon\)" and the odds are that they will respond “there is a \(\delta\)..."

Connection between limits of sequences and limits of functions

Here’s an equivalent way to define limits of functions: the limit \(\lim_{z \to z_0} f(z) = a\) if, for every sequence of points \(\{z_n\}\) with limit \(z_0\) the sequence \(\{f(z_n)\}\) has limit \(a\).