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 [latex]{\bf{r}}[/latex] approaches the limit [latex]\bf{L}[/latex] as [latex]t[/latex] approaches [latex]a[/latex], written
provided
Limit of a Vector-Valued Function Theorem
Let [latex]f[/latex], [latex]g[/latex], and [latex]h[/latex] be functions of [latex]t[/latex]. Then the limit of a vector-valued function [latex]{\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}[/latex] as [latex]t[/latex] approaches [latex]a[/latex] is given by
provided the limits [latex]\displaystyle\lim_{t{\to}a}f\,(t)[/latex] and [latex]\displaystyle\lim_{t{\to}a}g\,(t)[/latex] exist. Similarly, the limit of the vector-valued function [latex]{\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}+h\,(t)\,{\bf{k}}[/latex] as [latex]t[/latex] approaches [latex]a[/latex] is given by
provided the limits [latex]\displaystyle\lim_{t{\to}a}f\,(t),\ \displaystyle\lim_{t{\to}a}g\,(t)[/latex], and [latex]\displaystyle\lim_{t{\to}a}h\,(t)[/latex] 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 [latex]\displaystyle\lim_{t{\to}3}{\bf{r}}\,(t)[/latex] for
-
- a. [latex]{\bf{r}}\,(t)=(t^{2}-3t+4)\,{\bf{i}}+(4t+3)\,{\bf{j}}[/latex]
- b. [latex]{\bf{r}}\,(t)=\frac{2t-4}{t+1}\,{\bf{i}}+\frac{t}{t^{2}+1}\,{\bf{j}}+(4t-3)\,{\bf{k}}[/latex]
Try It
Calculate [latex]\displaystyle\lim_{t{\to}-2}{\bf{r}}\,(t)[/latex] for the function [latex]{\bf{r}}\,(t)=\sqrt{t^{2}-3t-1}\,{\bf{i}}+(4t+3)\,{\bf{j}}+\sin{\frac{(t+1)\,\pi}{2}}\,{\bf{k}}[/latex].
Watch the following video to see the worked solution to the above Try It
Definition
Let f, g, and h be functions of t. Then, the vector-valued function [latex]{\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}[/latex] is continuous at point [latex]t=a[/latex] if the following three conditions hold:
- [latex]{\bf{r}}\,(a)[/latex] exists
- [latex]\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)[/latex] exists
- [latex]\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)={\bf{r}}\,(a)[/latex]
Similarly, the vector-valued function [latex]{\bf{r}}\,(t)=f\,(t)\,{\bf{i}}+g\,(t)\,{\bf{j}}+h\,(t)\,{\bf{k}}[/latex] is continuous at point [latex]t=a[/latex] if the following three conditions hold:
- [latex]{\bf{r}}\,(a)[/latex] exists
- [latex]\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)[/latex] exists
- [latex]\lim_{t{\to}a}{\bf{r}}\,(t)={\bf{r}}\,(a)[/latex]