Limits Formalized Definition

Formalized Definition

The limit ${\lim }_{x\to c}=L$ (where there is an implicit assumption that $\exists L\in R$) Such that $\forall ϵ>0,\exists \delta >0\text{ s.t }\forall x\in \mathbb{R},0<|x-c|<\delta ⟹|f(x)-L|<ϵ$

Non-Formal Definition

Think of it like a shrinking windows around point $c$ This means that no matter what window of $L\pm ϵ$, you should be able to find values of $c$ within the constraints of $c\pm \delta$ If this is true for every epsilon, then the limit exists.

Limit Definition Cases

Example with Infinities

If, for example we say ${\lim }_{x\to \infty }$, then we can say $x>N$ where $N>0$ and $N$ is a represents a big number. If, for example we say ${\lim }_{x\to -\infty }$, then we can say $x<N$ where $N<0$ and $N$ is a represents a small number. Such is the same for $f(x)=\infty$

Formalized Range Biconditional

$|x-c|<\delta$ $\iff -\delta <x-c<\delta$ $\iff -\delta +c\le x\le c+\delta$ $⟺x\in (c-\delta ,c+\delta )$