Im studying calculus this year and one of the lectures included undefined values (*forbidden and unwanted* not my words btw). These included:
1. 0/0
2. oo – oo
3. oo/oo
4. 0*oo
5. 0^0
6. oo^0
7. 1^oo
All of these are extremely weird to me and I don’t really understand them, but the one that strikes me the most is the last one. As a former math competitor and regarded as “gifted” in math, I feel stupid not being able to comprehend this, but most importantly it shatters my belief that math can explain everything and that is has all the answers.
I don’t see infinity as a really big number, I understand it as a concept, and what confuses me the most is seeing infinity treated as number. 1^oo for me doesn’t make any sense. It seems mathematically absurd. Infinity isn’t number, it’s a quantity that can’t be measured. And if it is treated as a number shouldn’t 1^oo = 1?
How come these are “undefined”? Someone please answer, Im losing my mind over this. All explanations are welcome, shallow or deep.
edit: to clarify oo = infinity
In: Mathematics
To be more precise, we’re actually talking about the limit of 1^n as n approaches infinity.
Perhaps the easiest way to understand this would be to consider the concept of a left-hand and right-hand limit. That is, we choose something that bounds our limit arbitrarily close on either side.
If you’ve got the limit of 2^n as n approaches infinity, it’s pretty easy. The limit would be bounded by (2-d)^n and (2+d)^n, where d is some arbitrarily small number – and both of those evaluate to the same limit as 2^n itself for very small values of d.
But what happens when we use (1-d)^n and (1+d)^n ? In the first case, our limit is zero (no matter how small d may be). In the second case, our limit is infinite (again, no matter how small d may be). So all we can really say is that the limit of 1^n as n goes to infinity is somewhere between zero and infinite.
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