[[!redirects algebraic limit theorem]] #Contents# * table of contents {:toc} ## Idea ## A general structure where the various concepts in scalar calculus makes sense. ## Definition ## Let $F$ be a [[Heyting field]] and a [[function limit space]], where $x^{-1}$ is another notation for $\frac{1}{x}$. $F$ is a __calculus field__ if the __algebraic limit theorems__ are satisfied, i.e. if the limit preserves the field operations: $$z:\prod_{c:S} \left(\lim_{x \to c} 0(x) = 0\right)$$ $$a:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \prod_{g:FuncWithLim(S, F)(c)} \left(\lim_{x \to c} f(x) + g(x) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)\right)$$ $$n:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \left(\lim_{x \to c} -f(x) = -\lim_{x \to c} f(x) \right)$$ $$s:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \prod_{g:FuncWithLim(S, F)(c)} \left(\lim_{x \to c} f(x) - g(x) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)\right)$$ $$\alpha:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \prod_{a:\mathbb{Z}} \left(\lim_{x \to c} a f(x) = a \left(\lim_{x \to c} f(x)\right) \right)$$ $$\alpha:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \prod_{a:S} \left(\lim_{x \to c} a f(x) = a \left(\lim_{x \to c} f(x)\right) \right)$$ $$o:\prod_{c:S} \left(\lim_{x \to c} 1(x) = 1\right)$$ $$m:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \prod_{g:FuncWithLim(S, F)(c)} \left(\lim_{x \to c} f(x) \cdot g(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)\right)$$ $$p:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \prod_{n:\mathbb{N}} \left(\lim_{x \to c} {f(x)}^n = {\left(\lim_{x \to c} f(x)\right)}^n \right)$$ $$r:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \left(\left(\lim_{x \to c} g(x)\right) \# 0\right) \to \left(\lim_{x \to c} {f(x)}^{-1} = {\left(\lim_{x \to c} f(x)\right)}^{-1}\right)$$ $$i:\prod_{c:S} \prod_{f:FuncWithLim(S, F)(c)} \left(\left(\lim_{x \to c} x\right) \# 0\right) \to \left(\lim_{x \to c} f(x) \cdot {f(x)}^{-1} = 1\right)$$ ## See also ## * [[Newton-Leibniz operator]]