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).