Introduction to Vector-Valued Functions: Learn It 3

Limits and Continuity of a Vector-Valued Function

Understanding limits of vector-valued functions is essential for studying their calculus. The process builds directly on what you know about limits of single-variable functions.

limit of a vector valued function

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 the rigorous definition, but in practice we use a simpler approach: calculate the limit of each component function separately.

limit of a vector-valued function theorem

Let [latex]f[/latex], [latex]g[/latex], and [latex]h[/latex] be functions of [latex]t[/latex].

For plane curves:

[latex]\underset{t\to a}{\text{lim}}{\bf{r}}(t)=\bigg[\underset{t\to a}{\text{lim}}f(t)\bigg]{\bf{i}}+\bigg[\underset{t\to a}{\text{lim}}g(t)\bigg]{\bf{j}}[/latex]

provided the limits [latex]\underset{t\to a}{\text{lim}}f(t)[/latex] and [latex]\underset{t\to a}{\text{lim}}g(t)[/latex] exist.

For space curves:

[latex]\underset{t\to a}{\text{lim}}{\bf{r}}(t)=\bigg[\underset{t\to a}{\text{lim}}f(t)\bigg]{\bf{i}}+\bigg[\underset{t\to a}{\text{lim}}g(t)\bigg]{\bf{j}}+\bigg[\underset{t\to a}{\text{lim}}h(t)\bigg]{\bf{k}}[/latex]

provided the limits [latex]\underset{t\to a}{\text{lim}}f(t)[/latex], [latex]\underset{t\to a}{\text{lim}}g(t)[/latex], and [latex]\underset{t\to a}{\text{lim}}h(t)[/latex] exist.

Think of this as taking the limit “component by component.” Each component function follows the same limit rules you learned in Calculus 1.

In the following example, we show how to calculate 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

Now that we can calculate limits of vector-valued functions, we can define what it means for such a function to be continuous.

continuity of a vector-valued function

Let [latex]f[/latex], [latex]g[/latex], and [latex]h[/latex] be functions of [latex]t[/latex].

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 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, [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 same three conditions hold.

These three conditions should look familiar—they’re the same requirements for continuity you learned with single-variable functions. A vector-valued function is continuous at a point when you can draw its graph through that point without lifting your pencil.