Negative infinity
We don't know what infinity means. That seems strange, but nobody really can imagine what it is. Mathematicians can live with that. Calculations with positive infinity is possible, and calculations with negative infinity follow the same pattern. But pay attention: infinity is not a number.
Explanation
First we make an addition
That is the same for every number, even if it is a very large number. As a consequence
− ∞ + ∞ = ?
And that is clear, as infinity has no fixed value. Now look at a subtraction
and also
A multiplication is also possible. We use a positive constant c and get
Of course
whatever that may be. Now look at a division. That is always a bit tricky, so here you get
That is not a big surprise. However
?
We can accept that. When this calculation emerges in an other way, you may get an answer, as
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Here you use n→−∞ what means: It moves toward negative infinity, but is still a number. In the numerator and the denominator it concerns the same (infinit) value, and therefor you can perform the division. And finally
−∞ × 0 = ?
This is quite straight forward. The number 0 may always cause special problems, and ∞ is no number. We could expect this. For the sake of completeness also the division by infinitely small, what we write here as
Here you work with Δx→0− and it means: it is approaching from the negative to 0, but is not 0, and you may therefore divide by it. You can always make it even crazier, and then you write something like
,
,
and so on.
That's kind of fun, but we will not look into it.