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But here's your problem: infinity * n = infinity, right? 0 * n = 0, right? 0 * infinity = ? Ok, this is relevant here particularly because: Lim 1/x as x ->

But here's your problem: infinity * n = infinity, right? 0 * n = 0, right? 0 * infinity = ? Ok, this is relevant here particularly because: 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.

Twisol 13y ago
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. A quote of Knuth, taken from Wikipedia (http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_powe...): > 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."
- einhverfr 13y ago
  Regarding Knuth's point, that's a good one, since f(x) and g(x) are not necessarily the same.

Twisol 13y ago

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. A quote of Knuth, taken from Wikipedia (http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_powe...): > 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."

einhverfr 13y ago
Regarding Knuth's point, that's a good one, since f(x) and g(x) are not necessarily the same.