I prefer thinking about 0^0 = 1 as an empty product (https://en.wikipedia.org/wiki/Empty_product), since it generalizes nicely to any operation with an identity element. That is, if you apply any operation zero times, the result is that operation's identity. It's interesting that the analogous empty sum, 0*1 = 0, is a complete non-issue.
Lim 1/x as x -> 0 from the positive side is infinity, right?
So 0 * that is.....
lim 1/x as x -> from the negative side is negative infinity, right?
So 0 * that is.....
That's why 0/0 doesn't work as such. You don't know how 0 is derived or what it means. If we have x^2/x, and take the limit as x -> 0, we get 0. If we take x/x^2 and take the limit as x -> 0 we get +/- infinity.
But that doesn't mean that x/x or x^2/x are not continuous functions, any more than 1 or x are not continuous functions.
You are conflating the limiting behavior of a function with the value of a function. In (standard) analysis, there is no actual value called infinity - it's just used to describe how a function behaves arbitrarily close to a given value.
When you have a 0^0 limiting form (or 0/0, or 0 * infinity), the function's behavior is indeterminate. The "0" and "infinity" you're looking at aren't precisely 0 or infinity, but only arbitrarily close. The actual behavior of the function depends on the expressions that approach 0 and infinity, hence `x/(x+1)` and `x/e^x` having different limiting behaviors as x increases without bound.
But if you are literally considering the function at a specific point which produces a so-called indeterminate form, the answer is simpler. In some cases, the function is undefined (e.g. 1/x where x=0). In the case of 0^0, there is a precise value we can assign: 1. And this doesn't conflict with the 0^0 limiting form: the limit of a function at a point can be different from the actual value of the function at said point.
> Knuth (1992) contends strongly that 0^0 "has to be 1", drawing a distinction between the value 0^0, which should equal 1 as advocated by Libri, and the limiting form 0^0 (an abbreviation for a limit of f(x)^g(x) where f(x), g(x) -> 0), which is necessarily an indeterminate form as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."
