From 0eed6a8a93cc86e0f6158e43e421551814d97680 Mon Sep 17 00:00:00 2001 From: Luc Date: Fri, 10 Nov 2023 10:32:05 +0100 Subject: [PATCH] Updated formal definitions of the limit. --- docs/en/mathematics/calculus/limits.md | 23 ++++++++++++++++++++--- 1 file changed, 20 insertions(+), 3 deletions(-) diff --git a/docs/en/mathematics/calculus/limits.md b/docs/en/mathematics/calculus/limits.md index 61a6e4b..ef40182 100755 --- a/docs/en/mathematics/calculus/limits.md +++ b/docs/en/mathematics/calculus/limits.md @@ -44,11 +44,28 @@ If $\lim_{x \to a} f(x) = L$, $\lim_{x \to a} g(x) = M$, and $k$ is a constant t The limit $\lim_{x \to a} f(x) = L$ means, $$ -\forall \varepsilon > 0, \exists \delta \space \mathrm{,s.t.,} \space -\forall x \in \mathbb{R}, \space 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon +\forall \varepsilon_{> 0} \exists \delta_{>0} \Big[ 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon \Big]. $$ -For one-sided, infinite and limits at infinity there are similar formal definitions. +The limit $\lim_{x \to \infty} f(x) = L$ means, + +$$ +\forall \varepsilon_{> 0} \exists N_{>0} \Big[x > N \implies |f(x) - L | < \varepsilon \Big]. +$$ + +The limit $\lim_{x \to a} f(x) = \infty$ means, + +$$ +\forall M_{> 0} \exists \delta_{>0} \Big[ 0<|x-a|<\delta \implies f(x) > M \Big]. +$$ + +The limit $\lim_{x \to \infty} f(x) = \infty$ means, + +$$ +\forall M_{> 0} \exists N_{>0} \Big[ x > N \implies f(x) > M \Big]. +$$ + +For one-sided limits there are similar formal definitions. ### Example