Limit of a function of several variables

Definition

Let $D \subseteq \mathbb{R}^n$ and let $f \colon D \to \mathbb{R}^m$. Let $a \in \mathbb{R}^n$ be a limit point of $D$: each open ball about $a$ contains a point of $D$ other than $a$. Let $L \in \mathbb{R}^m$. When this holds, $f(x)$ tends to $L$ as $x$ tends to $a$, with the notation $\lim_{x \to a} f(x) = L$. For each $\varepsilon > 0$ there is a $\delta > 0$ such that $\|f(x) - L\| < \varepsilon$ for each $x \in D$ with $0 < \|x - a\| < \delta$. Here $\|\cdot\|$ is the Euclidean norm of Euclidean space.

Remarks

A limit is unique when it exists, because $a$ is a limit point of $D$. The value $f(a)$, and even whether $a$ is in $D$, has no effect on the limit. A function into $\mathbb{R}^m$ tends to $L$ if and only if each component $f_i$ tends to $L_i$. To show that a limit does not exist, it is enough to find two paths with different limits.

Explanations

L1 human

The definition says that you can make $f(x)$ as close to $L$ as you want. For this, take $x$ close enough to $a$ but not equal to $a$. In the plane, "close to $a$" means inside a small disc about $a$, so $x$ can come from every direction at once.