91Ó°ÊÓ

To prove,

$\underset{t→t_0}{lim} \left(c_1{\mathbf{r}}_1\left(t\right)+c_2{\mathbf{r}}_2\left(t\right)\right)=c_1{\mathbf{r}}_1\left(t_0\right)+c_2{\mathbf{r}}_2\left(t_0\right)$

Consider two continuous vector functions with two components.

Let

The objective is to prove that

Since $r_1\left(t\right)and{r}_2\left(t\right)$are continuous, their components $x_1\left(t\right),y_1\left(t\right),x_2\left(t\right),y_2\left(t\right)$are continuous. The product and sum of the continuous functions are also continuous.

$\left(c_1{\mathbf{r}}_1\left(t\right)+c_2{\mathbf{r}}_2\left(t\right)\right)=c_1{\mathbf{r}}_1\left(t_0\right)+c_2{\mathbf{r}}_2\left(t_0\right)$

Therefore, $c_1{r}_1\left(t\right)+c_2{r}_2\left(t\right)$is a continuous function.

Consider

Thus,$\underset{t→t_0}{lim} \left(c_1{\mathbf{r}}_1\left(t\right)+c_2{\mathbf{r}}_2\left(t\right)\right)=c_1{\mathbf{r}}_1\left(t_0\right)+c_2{\mathbf{r}}_2\left(t_0\right)$