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.