First look at limits

Now that we have a good understanding of sequences and of induction theorem, we can start to study limits. Hopefully understanding limits in a the case of sequences will help us understand limits in the case of functions. First we will look at concepts with vague definitions, and then we can move to a more formal definition. Using the formal and exact definition of limits will help us really understand this concept.

Definition

For a sequence, we mostly want to figure out the behavior of the sequence when $n$ becomes very very large. Let’s suppose that a sequence $a_n$ is definined for all positive integers $n$. We have four cases:

1. The sequence values can go as high as we want, in this case we say that the sequence diverges to $+\infty$.
2. The sequence values can go as low as we want, in this case we say that the sequence diverges to $-\infty$.
3. The sequence values can get very close to a certain value $L$ (an extreme case is when the sequence is actually equal to $L$ after a certain value). In this case we say that the sequence converges to $L$.
4. All the other cases (for example, the sequence oscillates between two values), where we can not say anything about the behavior of the sequence.

Examples

• Arithmetic sequences diverge to $+\infty$ or $-\infty$ depending on the sign of $d$.
• For geometric sequences, if $|r|< 1$, then the sequence converges to $0$. If $r> 1$, then the sequence diverges to $+\infty$ or $-\infty$ depending on the sign of $r$. But if $r< -1$, then the sequence doesn’t have any limits. We can see that in our graphs