極限の定義

数列の極限

$(a_i)_{i\in\mathbb{N}}$ where $a_i\in\mathbb{R}$ then
$\lim_{i\rightarrow\infty} a_i = a (\in\mathbb{R})$ is defined below
$$ \forall \epsilon \gt 0, \exists n_0 \in \mathbb{N}, n_0 \lt n \Rightarrow |a_n - a| < \epsilon $$
$\lim_{i\rightarrow\infty} a_i = \infty$ is defined below
$$ \forall M \gt 0, \exists n_0 \in \mathbb{N}, n_0 \lt n \Rightarrow M \lt a_n $$
$\lim_{i\rightarrow\infty} a_i = -\infty$ is defined below
$$ \forall m \lt 0, \exists n_0 \in \mathbb{N}, n_0 \lt n \Rightarrow a_n \lt m $$

関数の極限

$$A\subset\mathbb{R}$$ $$f:A\rightarrow\mathbb{R}$$ then $$ \lim_{x\rightarrow a} f(x) = b$$ defained by $$(\forall \epsilon \gt 0)(\exists \delta \gt 0)(\forall x \in A)(|x - a| \lt \delta \Longrightarrow |f(x) - b| \lt \epsilon)$$