Answer
$$\eqalign{
& \left( {\text{a}} \right){\text{ Continuous}} \cr
& \left( {\text{b}} \right){\text{ Not continuous}} \cr} $$
Work Step by Step
$$\eqalign{
& \left( {\text{a}} \right){\text{ }}{\bf{r}}\left( t \right) = 3\sin t{\bf{i}} - 2t{\bf{j}} \cr
& {\text{Calculating if it is continuous at }}t = 0, \cr
& \mathop {\lim }\limits_{t \to 0} {\bf{r}}\left( t \right) = \mathop {\lim }\limits_{t \to 0} \left( {3\sin t{\bf{i}} - 2t{\bf{j}}} \right) \cr
& {\text{ }} = {\bf{i}}\mathop {\lim }\limits_{t \to 0} 3\sin t - {\bf{j}}\mathop {\lim }\limits_{t \to 0} 2t \cr
& {\text{ }} = {\bf{i}}\left( {3\sin \left( 0 \right)} \right) - {\bf{j}}2\left( 0 \right) \cr
& {\text{ }} = 0 \cr
& {\text{Therefore, it is cointinuous at }}t = 0 \cr
& \left( {\text{b}} \right){\text{ }}{\bf{r}}\left( t \right) = {t^2}{\bf{i}} + \frac{1}{t}{\bf{j}} + t{\bf{k}} \cr
& \mathop {\lim }\limits_{t \to 0} {\bf{r}}\left( t \right) = \mathop {\lim }\limits_{t \to 0} \left( {{t^2}{\bf{i}} + \frac{1}{t}{\bf{j}} + t{\bf{k}}} \right) \cr
& {\text{ }} = {\bf{i}}\mathop {\lim }\limits_{t \to 0} {t^2} + {\bf{j}}\mathop {\lim }\limits_{t \to 0} \frac{1}{t} + {\bf{k}}\mathop {\lim }\limits_{t \to 0} t \cr
& {\text{ }} = {\bf{i}}{\left( 0 \right)^2} + {\bf{j}}\frac{1}{0} + {\bf{k}}\left( 0 \right) \cr
& \mathop {\lim }\limits_{t \to 0} \frac{1}{t},{\text{ does not exist, then }}{\bf{r}}\left( t \right){\text{is not continuous at }}t = 0 \cr} $$
