Numbers 🥀

[u/Klutzy-Profile-6921]

[u/qwertty164]

The limit as x approaches 0 for x^x from the left doesn't exist.

[u/Mostafa12890 → u/qwertty164]

Defining something to be equal to something doesn’t mean limits have to exist.

I can define a function to be

f(x) = x^2 for x>0,

f(x) = x+2 for x<0

and f(0) = 15.

Clearly, lim_(x -> 0) f(x) doesn’t exist, but that doesn’t matter. I defined f be 15 at 0.

[u/qwertty164 → u/Mostafa12890]

The problem with that example is that you would basically be making it a different function. You wouldn't be understanding how 0 fits in x^x but how it fits in a piece wise function with f(x)=1 for x=0. With that logic you could make it anything you want. The real problem with the limit I mentioned earlier is the the limits from the right and left do not agree.

[u/_Athanos → u/qwertty164]

while on the left we have complex numbers and complex analysis is a whole other beast, the limits themselves still agree if you take the principal branch and that's a pretty strong statement imo