This is also why L'Hôpital's rule is necessary for finding some limits in calculus.

which is awesome

(but sadly rarely usable on complex cases )

These
(see attached : it's a math homework about calculating limits of the functions written)


(maybe some of them are doable, but probably not all)
(since you can't always take the derivative (easily))

Well it seems that all of the fancy and complex responses have been taken... The simplest way it was described to me was that you can't take nothing from nothing, because there's nothing to give and you're really not getting anything in return.

inb4 someone finds some logical fallacy with that.

I understand what you're saying. I learn about series next year.

Wolfram Alpha is great but it feels frustrating that it now costs money to use all of its cool new features (and a couple of its old features).
Since it's still getting new things added to it, it's fine that it doesn't always show exactly what I want, yet. (I'll wait until it can read minds.)

The interesting thing is when you start working with limits.  Lets say you have two equations that both equal zero at X=1, and lets say we divide one equation by the other.  If we find the ratio at X=1, obviously that is undefined.  But if we get reeeeallly close to X=1 (take the limit as X->1) the ratio can actually approach any number!  Depending on the equation of course, I once devised an equation that was undefined at X=0, but the limit approached 42

Yeah, we're learning Calculus (mostly differential atm) in Year 12, and we had a long discussion about limits.
The limit technically does not exist at the given value e.g.: lim h->0 f(x)=3x^2+7x-5.
At h=0 the equation is undefined as we're dividing by 0 (The answer could be infinity, for all we know, or it could be 9000.00000032), but the point of Calculus (well at least this part of calculus) is that we are to find the theoretical derivative of the equation when h=0, even though it technically does not exist.

Hope that made some sense, and I may have made a mistake.

Think about this....

Take a number, n. Subtract 0 from it and keep going until n=0. Let me know when you're done.

If you think about it that way, you get infinity. Technically, infinity IS an undefined number (the textbook definition of infinity is the inability to assign a definite quantity).