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
I found [this answer](https://math.stackexchange.com/a/10493) on StackExchange that explains it quite elegantly.
To summarize, the actual fixed expression 1^∞ *will* converge to 1 if taken as lim x-> ∞ (1^(x)). But the thing the math book is warning you about are expressions that contain multiple parts that look like they would *become* 1^∞ when their limits are taken in pieces.
One of the examples given is this expression:
(1 + 1/n)^n
What’s the limit of this if n -> ∞? If you take the part inside the parenthesis on its own, the fraction tends to 1/∞ = 0, so the expression becomes 1. Taking this entire thing to the *n*th power as n->∞ gives us the form 1^∞ . Does this equal one? No. In fact, the limit of this expression as n->∞ is the constant *e*, of all things! Weird, right?
The thing is, if you have two functions, f(x) and g(x), composed together like f(x)^g(x) , where x->∞ leads to f(x) -> 1 and g(x) -> ∞, the limit of the composition could be *any number of things*. It all depends on what f(x) and g(x) actually are. They all *look* like they go to 1^∞, but they all behave differently. That is why 1^∞ is indeterminate form.
To think about it another way, if you have an expression that can be split into two pieces, f(x) and g(x), those two pieces can approach their limits at different “speeds”. For most limits, this effect doesn’t really matter, one of those pieces will likely be in a position where its “speed” is so much faster than the other that the effect of the other function gets completely nullified. For these, you’re allowed to take “naive” limits (sub in ∞ for all the variables and do “algebra” with it, treating divisions by ∞ as zero) of expressions. But if we get constructs where those speeds actually matter, the final limit could end up being basically anything depending on what those speeds actually are relative to one another. So naive approaches to evaluating limits that completely disregard this kind of nuance will almost always fail to arrive at the correct answer. Any expression that reduces to one of the “indeterminate forms” after a naive limit is taken is one of these expressions. This list of forms is warning you that, “hey, the speeds actually matter here, we need to use an approach that takes this into account”. Methods like L’hopital’s rule will allow you to do this for some expressions. That’s what all those funny calculus rules are for, they’re special methods that take these fighting “speeds” into account when calculating the limit. Any expression that becomes 1^∞ after one of these naive limits happens to fall into this territory, needing a special rule to calculate correctly.
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