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 [latex]{\bf{r}}[/latex] approaches the limit [latex]\bf{L}[/latex] as [latex]t[/latex] approaches [latex]a[/latex], written

[latex]\displaystyle{\lim_{t\to{a}}}\;{\bf{r}}\,(t)={\bf{L}}[/latex],

 

provided

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

 

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 [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

[latex]\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}}[/latex],

 

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

[latex]\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}}[/latex]

 

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

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

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 [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:

  1. [latex]{\bf{r}}\,(a)[/latex] exists
  2. [latex]\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)[/latex] exists
  3. [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:

  1. [latex]{\bf{r}}\,(a)[/latex] exists
  2. [latex]\displaystyle\lim_{t{\to}a}{\bf{r}}\,(t)[/latex] exists
  3. [latex]\lim_{t{\to}a}{\bf{r}}\,(t)={\bf{r}}\,(a)[/latex]