[Math] Limit of a multi-variate function and Limit Point

Definition of Limit [= ($\epsilon$-$\delta$) definition] 

Assume that $S\subseteq \mathbb{R}^n$, and that $f:S\to \mathbb{R}$ is a mutivariate function. The statement

$$ \underset{\textbf{x} \to \textbf{a}}{\lim} f (\textbf{x})= L$$

is defined to mean that

$$\begin{equation}\label{lim.def}
\forall \varepsilon >0, \ \ \exists \delta>0 \quad\mbox{ such that if } \mathbf x \in S \mbox{ and } 0 < || \mathbf x - \mathbf a ||_2<\delta, \mbox{ then } |\mathbf f(\mathbf x) - {L}| < \varepsilon.
\end{equation}$$

 

위 정의는,

• 아주 작은 값의 $\epsilon$이 주어지더라도 (일반적으로 $\epsilon$은 0에 가까운 매우 작은 값으로 0과 구분이 거의 안되는 수준의 값임),
• $\textbf{x}$와 $\textbf{a}$의 difference vector의 L-2 norm을 $\delta$보다 작게할 경우 (=적절한 수준 이하로 작게 처리),
• $f(\textbf{x})$와 $L$의 차이가 $\epsilon$보다 작아지게 할 수 있음 (=거의 같게 만들 수 있음)을 의미함.

https://bme808.blogspot.com/2022/10/norm.html

Norm (노름)

 

In order for the definition to make sense,

we need to assume that $\textbf{a}$ is a limit point of $S$

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Definition of Limit point

The point $\textbf{a} \in \mathbb{R}^n$ is a limit point of set $S$ if and only if

$$\begin{equation}\label{limitpoint} \forall \delta>0, \quad
\exists \mathbf x\in S \quad\mbox{ such that }\quad
0 < |\mathbf x - \mathbf a|<\delta.
\end{equation}$$

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Example

다음 그림은 2개의 독립변수를 가지는 vector field에서 limit을 그림으로 보여준다.

ref. : https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/12%3A_Functions_of_Several_Variables/12.02%3A_Limits_and_Continuity_of_Multivariable_Functions

• iIlustrating the definition of a limit.
• The open disk in the x-y plane has radius $\delta$.
• Let $(x,y)$ be any point in this disk; $f(x,y)$ is within $\epsilon$ of $L$.

참고로, 3개의 독립변수인 경우($n=3$)엔 $S$가 ball 이 된다. 

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Reference

2024.02.27 - [.../Math] - [Math] Limit of Scalar Function: Left-sided Limit and Right-sided Limit

[Math] Limit of Scalar Function: Left-sided Limit and Right-sided Limit

http://www.math.toronto.edu/courses/mat237y1/20199/notes/Chapter1/S1.2.html

1.2: Limits and Continuity

https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/12%3A_Functions_of_Several_Variables/12.02%3A_Limits_and_Continuity_of_Multivariable_Functions 

12.2: Limits and Continuity of Multivariable Functions