Let $r_1\left(t\right)and{r}_2\left(t\right)$be continuous vector functions with three components, and let $t_0$be a point in the domains of both $r_{1}and{r}_2$. Prove that

localid="1649617482201" $\underset{t\to t_0}{lim} \left({\mathbf{r}}_1\left(t\right)\times {\mathbf{r}}_2\left(t\right)\right)={\mathbf{r}}_1\left(t_0\right)\times {\mathbf{r}}_2\left(t_0\right)$

given,

$\underset{t\to t_0}{lim} \left({\mathbf{r}}_1\left(t\right)\times {\mathbf{r}}_2\left(t\right)\right)={\mathbf{r}}_1\left(t_0\right)\times {\mathbf{r}}_2\left(t_0\right)$

Consider two continuous vector functions with three components. Let

The objective is to prove that

$\underset{t\to t_0}{lim} \left({\mathbf{r}}_1\left(t\right)\times {\mathbf{r}}_2\left(t\right)\right)={\mathbf{r}}_1\left(t_0\right)\times {\mathbf{r}}_2\left(t_0\right)$where $t_0$belongs to the domains of $r_{1}and{r}_2$.

Since role="math" localid="1649617811675" $r_1\left(t\right)and{r}_2\left(t\right)$are continuous, their components $x_1\left(t\right),y_1\left(t\right),z_1\left(t\right),x_2\left(t\right),y_2\left(t\right),z_2\left(t\right)$are continuous. Thus product and sum of the continuous functions are also continuous.

Therefore,$r_1\left(t\right)\times r_2\left(t\right)$is continuous.

From the definition of a continuous function.

Thus,$\underset{t\to t_0}{lim} \left({\mathbf{r}}_1\left(t\right)\times {\mathbf{r}}_2\left(t\right)\right)={\mathbf{r}}_1\left(t_0\right)\times {\mathbf{r}}_2\left(t_0\right)$