Limits and Continuity | Sahithyan's Notes

Let $\boldsymbol{r}(t)$ be defined for each $t$ in some open interval containing $a$ , except possibly at $a$ , and let $L$ be a vector. Then

\lim_{t \to a} \boldsymbol{r}(t) = L \quad \iff \quad \lim_{t \to a} \lVert \boldsymbol{r}(t) - L \rVert = 0

Properties

Limit on Component Functions

If $\boldsymbol{r}(t) = \langle x(t), y(t), z(t) \rangle$ and the limits of the component functions exist as $t \to a$ ,

\lim_{t \to a} \boldsymbol{r}(t) = \left\langle \lim_{t \to a} x(t), \lim_{t \to a} y(t), \lim_{t \to a} z(t) \right\rangle

For vector-valued functions $F(t)$ , $G(t)$ and real-valued function $h(t)$ with $\lim\limits_{t \to a} F(t)$ , $\lim\limits_{t \to a} G(t)$ , $\lim\limits_{t \to a} h(t)$ existing:

$\lim\limits_{t \to a} [F(t) \pm G(t)] = \lim\limits_{t \to a} F(t) \pm \lim\limits_{t \to a} G(t)$
$\lim\limits_{t \to a} [h(t)F(t)] = \left( \lim\limits_{t \to a} h(t) \right) \left( \lim\limits_{t \to a} F(t) \right)$
$\lim\limits_{t \to a} [F(t) \cdot G(t)] = \lim\limits_{t \to a} F(t) \cdot \lim\limits_{t \to a} G(t)$
$\lim\limits_{t \to a} [F(t) \times G(t)] = \lim\limits_{t \to a} F(t) \times \lim\limits_{t \to a} G(t)$

Continuity

$\boldsymbol{r}(t)$ is continuous at $a$ iff $\lim\limits_{t \to a} \boldsymbol{r}(t) = \boldsymbol{r}(a)$ . It is continuous on an interval $I$ if it is continuous at each point of $I$ .

$\boldsymbol{r}(t)$ is continuous at $a$ iff each component function is continuous at $a$ .