Section 13.1: Vector Functions and Space Curves

The limit of a vector function r is defined by taking the limits of its component functions as follows.

1. If \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), then \[ \lim_{t \to a} \mathbf{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle \] provided the limits of the component functions exist.

If \(\lim_{t \to a} \mathbf{r}(t) = \mathbf{L}\), this definition is equivalent to saying that the length and direction of the vector \(\mathbf{r}(t)\) approach the length and direction of the vector L.

Equivalently, we could have used an \(\epsilon-\delta\) definition (see Exercise 54). Limits of vector functions obey the same rules as limits of real-valued functions (see Exercise 53).

EXAMPLE 2 Find \(\lim_{t \to 0} \mathbf{r}(t)\), where \(\mathbf{r}(t) = (1+t^3)\mathbf{i} + te^{-t}\mathbf{j} + \frac{\sin t}{t}\mathbf{k}\).

A vector function r is continuous at a if \[ \lim_{t \to a} \mathbf{r}(t) = \mathbf{r}(a) \] In view of Definition 1, we see that r is continuous at \(a\) if and only if its component functions \(f\), \(g\), and \(h\) are continuous at \(a\).