Limits and Continuity of a Vector-Valued Function

Learning Outcomes

Define the limit of a vector-valued function.

We now take a look at the limit of a vector-valued function. This is important to understand to study the calculus of vector-valued functions.

Definition

A vector-valued function ${\bf{r}}$ approaches the limit $\bf{L}$ as $t$ approaches $a$ , written

lim_{t \to a} r (t) = L

$\displaystyle{\lim_{t\to{a}}}\;{\bf{r}}\,(t)={\bf{L}}$ ,

provided

lim_{t \to a} ‖ r (t) - L ‖ = 0

$\displaystyle{\lim_{t\to{a}}}\left\|{\bf{r}}\,(t)-{\bf{L}}\right\|=0$ .

This is a rigorous definition of the limit of a vector-valued function. In practice, we use the following theorem:

Limit of a Vector-Valued Function Theorem

Let $f$ , $g$ , and $h$ be functions of $t$ . Then the limit of a vector-valued function ${\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}$ as $t$ approaches $a$ is given by

lim_{t \to a} r (t) = [lim_{t \to a} f (t)] i + [lim_{t \to a} g (t)] j

$\displaystyle\lim_{t\to a}{\bf{r}}\,(t)=\bigg[\displaystyle\lim_{t{\to}a}f\,(t)\bigg]\,{\bf{i}}+\bigg[\displaystyle\lim_{t{\to}a}g\,(t)\bigg]{\bf{j}}$ ,

provided the limits $\displaystyle\lim_{t{\to}a}f\,(t)$ and $\displaystyle\lim_{t{\to}a}g\,(t)$ exist. Similarly, the limit of the vector-valued function ${\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}+h\,(t)\,{\bf{k}}$ as $t$ approaches $a$ is given by

lim_{t \to a} r (t) = [lim_{t \to a} f (t)] i + [lim_{t \to a} g (t)] j + [lim_{t \to a} h (t)] k

$\displaystyle\lim_{t\to a}{\bf{r}}\,(t)=\bigg[\displaystyle\lim_{t{\to}a}f\,(t)\bigg]\,{\bf{i}}+\bigg[\displaystyle\lim_{t{\to}a}g\,(t)\bigg]{\bf{j}}+\bigg[\displaystyle\lim_{t{\to}a}h\,(t)\bigg]\,{\bf{k}}$

provided the limits $\displaystyle\lim_{t{\to}a}f\,(t),\ \displaystyle\lim_{t{\to}a}g\,(t)$ , and $\displaystyle\lim_{t{\to}a}h\,(t)$ exist.

In the following example, we show how to calculate the limit of a vector-valued function.

Example: Evaluating the limit of a vector-valued function

For each of the following vector-valued functions, calculate $\displaystyle\lim_{t{\to}3}{\bf{r}}\,(t)$ for

Show Solution

Use the Limit of a Vector-Valued Function theorem and substitute the value $t=3$ into the two component expression:
$\begin{array}{ccc}\hfill \displaystyle\lim_{t{\to}3}{\bf{r}}\,(t)&=\hfill&{\displaystyle\lim_{t{\to}3}\big[(t^{2}-3t+4)\,{\bf{i}}+(4t+3)\,{\bf{j}}\big]}\hfill \\ \hfill & =\hfill & {\bigg[\displaystyle\lim_{t{\to}3}{(t^{2}-3t+4)}\bigg]\,{\bf{i}}+\bigg[\displaystyle\lim_{t{\to}3}{(4t+3)}\bigg]\,{\bf{j}}} \hfill \\ \hfill & =\hfill & {4\,{\bf{i}}+15\,{\bf{j}}}\hfill \\ \hfill \end{array}$
Use the Limit of a Vector-Valued Function theorem and substitute the value $t=3$ into the three component expression:
$\begin{array}{ccc}\hfill \displaystyle\lim_{t{\to}3}{\bf{r}}\,(t)&=\hfill&{\displaystyle\lim_{t{\to}3}\bigg(\frac{2t-4}{t+1}\,{\bf{i}}+\frac{t}{t^{2}+1}\,{\bf{j}}+(4t-3){\bf{k}}\bigg)}\hfill \\ \hfill & =\hfill & {\bigg[\displaystyle\lim_{t{\to}3}{(\frac{2t-4}{t+1})}\bigg]\,{\bf{i}}+\bigg[\displaystyle\lim_{t{\to}3}{(\frac{t}{t^{2}+1})}\bigg]\,{\bf{j}}+\bigg[\displaystyle\lim_{t{\to}3}{(4t-3)}\bigg]\,{\bf{k}}} \hfill \\ \hfill & =\hfill & {\frac{1}{2}\,{\bf{i}}+\frac{3}{10}\,{\bf{j}}+9\,{\bf{k}}}\hfill \\ \hfill \end{array}$

Try It

Calculate $\displaystyle\lim_{t{\to}-2}{\bf{r}}\,(t)$ for the function ${\bf{r}}\,(t)=\sqrt{t^{2}-3t-1}\,{\bf{i}}+(4t+3)\,{\bf{j}}+\sin{\frac{(t+1)\,\pi}{2}}\,{\bf{k}}$ .

Show Solution

lim_{t \to - 2} r (t) = 3 i - 5 j - k

$\displaystyle\lim_{t{\to}-2}{\bf{r}}\,(t)=3\,{\bf{i}}-5\,{\bf{j}}-{\bf{k}}$

Watch the following video to see the worked solution to the above Try It

You can view the transcript for “CP 3.3” here (opens in new window).

Now that we know how to calculate the limit of a vector-valued function, we can define continuity at a point for such a function.

Definition

Let f, g, and h be functions of t. Then, the vector-valued function ${\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}$ is continuous at point $t=a$ if the following three conditions hold:

${\bf{r}}\,(a)$ exists
$\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)$ exists
$\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)={\bf{r}}\,(a)$

Similarly, the vector-valued function ${\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}+h\,(t)\,{\bf{k}}$ is continuous at point $t=a$ if the following three conditions hold:

${\bf{r}}\,(a)$ exists
$\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)$ exists
$\lim_{t{\to}a}{\bf{r}}\,(t)={\bf{r}}\,(a)$

Module 3: Vector-Valued Functions