Limit (Vector)

vector-calculus analysis

Definition

Limit (Vector)

Let $D\subseteq {\mathbb{R}}^n$ and $f:D\to \mathbb{R}$ be a scalar-valued function. The limit ${\lim }_{\mathbf{x}\to {\mathbf{x}}_0}f(\mathbf{x})$ is a real number $c$ if, for every $\epsilon >0$, there exists a $\delta >0$ such that for all $\mathbf{x}\in D$:

$$0<\Vert \mathbf{x}-\mathbf{x}\_0\Vert \_2<\delta ⟹|f(\mathbf{x})-c|<\epsilon$$

where $\Vert \cdot {\Vert }_2$ denotes the Euclidean norm. This condition state that $f(\mathbf{x})$ can be made arbitrarily close to $c$ by restricting $\mathbf{x}$ to a sufficiently small delta neighbourhood around ${\mathbf{x}}_0$(excluding ${\mathbf{x}}_0$ itself).

Properties

• Unique: If a limit exists, it is unique.
• Direction Independence: For the limit to exist, the function must approach the same value $c$ regardless of the path taken to reach ${\mathbf{x}}_0$. If two different paths yield different limits, the limit does not exist.
• Linearity: The limit of a sum/product of functions is the sum/product of their limits, provided they exist.