this post is a pretty good example of why mathematicians are concerned with rigor ("obsessed with pedantry" if you're feeling less generous). if you came up with this on your own then you should be proud because it's an interesting way to poke a hole in the math you've been taught.
the tools to see why this doesn't work don't come until you start learning calculus. depending on where you live this might be late in secondary school or early in college. i think the americans don't do calculus in secondary school unless they're in the "AP" and the british don't do calculus in secondary if they only do the GCSE but do do calculus in "A-Levels"? I don't know I didn't get a secondary school degree from either of these places. but I digress.
Part 1:
what you said is this:
there's some boilerplate stuff you were probably supposed to say about x being a real number strictly greater than 0 (think about what happens if x<0 or if x = 0), but let's ignore that for a second.
basically, what's wrong is the words "infinity-eth root" and "1 ^ infinity". these don't actually mean anything, because roots and exponentiation are designed to work on numbers and infinity isn't a number.
what infinity actually is should probably be left to some very dense textbooks (you can definite it via set cardinality or via compaction or via geometry or via ...), but for our purposes it suffices to say that it is less a number and more the idea of "something bigger than all the other things".
the fact that it's an idea means you can't do math on it - for our purposes saying "infinity plus one" makes as little sense as saying "yellow plus one".
because you're doing something that doesn't make sense - doing math treating infinity as a number - you get a result that doesn't make sense.
Part 2:
but i mentioned that you can kind of approach this in a way that makes more sense using calculus.
rephrasing "infinity-eth root equals 1" using the language of calculus would look something like this:
and that statement is 100% correct. noone can really have any problems with it (besides the mathematicians who don't accept the existence of infinity but let's not divert ourselves too much here).
what this statement means is something along the lines of "let's look at what happens if you take the n-th root of x, where n actually is a number. as n gets bigger and bigger, the n-th root of x gets closer and closer to 1. if you let n get arbitrarily large, the result is arbitrarily close to 1."
we can take the rigorously defined mathematical form of the root and exponentiate that to the power of infinity in an equally rigorously defined way.
when we have an x, take the n-th root as n goes to infinity, and then exponentiate it by n as n goes to infinity, we just get x back out! everything works and we have a result that matches what you might expect. using the language of calculus, when you root x and then un-root the result you arrive back at x, instead of getting some weird contradiction.
we get a nice result in this scenario, but in general, the result of the expression "1^infinity" is actually an undefined value that can be whatever you want. in the example below, the stuff in the parentheses approaches 1, but the expression as a whole is equal to euler's constant, so around 2.72!
this is a very nice explanation as to why taking "1^infinity" is bad, and "Limits are about the journey not the destination" is a nice quote.
Part 3:
if you really want a tricky one try this out:
what!? i definitely wasn't able to figure out what was wrong with this one my first time seeing it... give it a spin yourself, but if you throw in the towel (like I did) then the top two explanations in
this link are both quite nice.
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TOO LONG DIDN'T READ SUMMARY:
infinity isn't a number and you can't use it like one or you'll be punished to an eternity of reading forum posts about basic calculus. wow react bellsprouts should be directed to the bottom right of this post.
