1 Infinite Limits

Before we cover any material from this week, its worth discussing the use of $\infty$ as a limit, especially when applying the Algebra of Limits. The main thing to note is that expressions such as $\infty - \infty$ and $\frac{\infty}{\infty}$ don’t really make a lot of sense, for example:

Example 1.1:

Give examples of sequences $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$, both diverging to $\infty$ for which:

1. $(a_n - b_n)_{n\in\mathbb{N}}$ diverges to $\infty$,
2. $(a_n - b_n)_{n\in\mathbb{N}}$ diverges to $-\infty$, and
3. $(a_n - b_n)_{n\in\mathbb{N}}$ converges to $0$.

Solution.

The idea here is to look for fairly simple sequences. With that in mind:

1. If we take $a_n = 2n$ and $b_n = n$, we see that $a_n - b_n = n$, and $(n)_{n\in\mathbb{N}}$ diverges to $\infty$. But if we tried to apply the algebra of limits to this result, it would suggest that $\infty - \infty = \infty$.
2. If we take $a_n = n$ and $b_n = 2n$, we see that $a_n - b_n = -n$, and $(-n)_{n\in\mathbb{N}}$ diverges to $-\infty$. But, again, if we tried to apply AoL to this result, it would suggest that $\infty - \infty = -\infty$. Immediately this conflicts with the answer to part 1)!
3. Finally, if we take $a_n = b_n = n$, we see that $a_n - b_n = 0$, and $(0)_{n \in \mathbb{N}}$ is a convergent sequence — it converges to $0$. Attempting to apply AoL to this result suggests that $\infty - \infty = 0$.

What parts 1), 2) and 3) demonstrate is that you can’t consistently define $\infty - \infty$. (Note: if you take the same sequences from 1), 2) and 3) and divide them, you can see why $\frac{\infty}{\infty}$ isn’t consistently define-able either).