In this module you learn different methods to calculate the limit of a number sequence.

Examine the following sequence: \[ 0, 0.3, 0.33, 0.333, 0.3333, 0.33333, \dots \] None of the elements in this sequence equals exactly \(\frac 13\), but we see that the values far away in the sequence lie very close to \(\tfrac 13\). The value \(\tfrac 13\) is called the limit of this sequence.

More generally, we say a number \(L\) is the limit of a sequence \(a_1, a_2, a_3, \dots\) if \(a_n\) tends to \(L\) as \(n\) becomes large. We write this as \[a_n\to L \text{ as } n\to\infty \] or alternatively as \[\lim_{n\to\infty} a_n = L\text.\] We also say that the sequence \(a_n\) converges to \(L\). Not every sequence has a limit and we will discuss this in more detail later.

Before looking at the precise mathematical definition of a limit we start off with an exercise.