Limits of Vector Functions

Limits

The limit of a vector function is defined as:

$$\begin{array}{rl}\underset{t\to a}{\lim }\vec{r}(t)& =\underset{t\to a}{\lim }⟨f(t),g(t),h(t)⟩\\ & \\ & =⟨\underset{t\to a}{\lim }f(t),\underset{t\to a}{\lim }g(t),\underset{t\to a}{\lim }h(t)⟩\\ & \\ & =\underset{t\to a}{\lim }f(t)\vec{i}+\underset{t\to a}{\lim }g(t)\vec{j}+\underset{t\to a}{\lim }h(t)\vec{k}\end{array}$$

Continuity

A vector valued function $\vec{v}(t)$ is called continuous at a point $t_0$ if

$$\vec{v}(t_0)=\underset{t\to t_0}{\lim }\vec{v}(t)$$

A vector valued function is continuous at a point if and only if its components are continuous at that point.