Limits of Real Vector Functions

Definition: Limit of a Real Vector Function

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ be a real vector function and let $\mathbf{a}\in {\mathbb{R}}^m$ be an Accumulation Point of $\mathcal{D}$.

We say that $\mathbf{L}\in {\mathbb{R}}^n$ is a limit of $f$ for $\mathbf{x}\to \mathbf{a}$ iff for each open ball $B_{\epsilon }(\mathbf{L})$ around $\mathbf{L}$ there exists an open ball $B_{\delta }(\mathbf{a})$ around $\mathbf{a}$ such that for all $\mathbf{x}\in \mathcal{D}$ different from $\mathbf{a}$,

$$\mathbf{x}\in B_{\delta }(\mathbf{a})⟹f(\mathbf{x})\in B_{\epsilon }(\mathbf{L})$$

Tip: Restatement without Open Balls

For the purposes of many proofs, it useful to define the limit without reference to open balls. This is done simply by taking the above definition and replacing every occurrence of “open ball” with the definition of an open ball.

We say that $\mathbf{L}\in {\mathbb{R}}^n$ is a limit of $f$ for $\mathbf{x}\to \mathbf{a}$ iff for each $\epsilon >0$, there exists some $\delta >0$ such that for all $\mathbf{x}\in \mathcal{D}$, if $0<||\mathbf{x}-\mathbf{a}||<\delta$, then $||f(\mathbf{x})-\mathbf{L}||<\epsilon$.

NOTATION

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

INTUITION

Suppose we have some fixed point $\mathbf{a}\in {\mathbb{R}}^m$ and a point $\mathbf{x}\in \mathcal{D}$ which we can move around freely. The limit ${\lim }_{\mathbf{x}\to \mathbf{a}}f(\mathbf{x})$ tells us what point in ${\mathbb{R}}^n$ (if any) $f(\mathbf{x})$ approaches as $\mathbf{x}$ gets closer and closer to $\mathbf{a}$. If the limit is $\mathbf{L}$, then no matter how small a sphere $B_{\epsilon }(\mathbf{L})$ we choose around $\mathbf{L}$, there will always be some sphere $B_{\delta }(\mathbf{a})$ (probably a very small one, too, but nevertheless still containing more than a single point) around $\mathbf{a}$ such that if $\mathbf{x}$ is inside $B_{\delta }(\mathbf{a})$, then $f(\mathbf{x})$ will be inside $B_{\epsilon }(\mathbf{L})$.

Theorem: Uniqueness of the Limit

If $f$ has a limit for $\mathbf{x}\to \mathbf{a}$, then this limit is unique.

PROOF

TODO

Characterizations

Theorem: Limit via Component Functions

Let $f:D\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ be a real vector function with component functions $f_1,\cdots ,f_n$ and let $\mathbf{a}\in {\mathbb{R}}^m$.

Then $f$ has a limit for $\mathbf{x}\to \mathbf{a}$ if and only if $f_1,\cdots ,f_n$ have limits for $\mathbf{x}\to \mathbf{a}$. Moreover,

$$\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})=\left[\begin{array}{c}\underset{\mathbf{x}\to \mathbf{a}}{\lim }{f}_1(\mathbf{x})\\ ⋮\\ \underset{\mathbf{x}\to \mathbf{a}}{\lim }{f}_n(\mathbf{x})\end{array}\right]$$

PROOF

TODO

Properties

Theorem: Linearity of the Limit Operator

If $f,g:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ have limits for $\mathbf{x}\to \mathbf{a}$, then for all $\lambda ,\mu \in \mathbb{R}$

$$\underset{\mathbf{x}\to \mathbf{a}}{\lim }[\lambda f(\mathbf{x})+\mu g(\mathbf{x})]=\lambda \underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})+\mu \underset{\mathbf{x}\to \mathbf{a}}{\lim }g(\mathbf{x})$$

PROOF

TODO