Answer
$$0{\bf{i}} + {\bf{j}}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{t \to {0^ + }} \left( {\sqrt t {\bf{i}} + \frac{{\sin t}}{t}{\bf{j}}} \right) \cr
& {\text{Evaluate the limit}} \cr
& \mathop {\lim }\limits_{t \to {0^ + }} \left( {\sqrt t {\bf{i}} + \frac{{\sin t}}{t}{\bf{j}}} \right) = \left( {\mathop {\lim }\limits_{t \to {0^ + }} \sqrt t {\bf{i}} + \mathop {\lim }\limits_{t \to {0^ + }} \frac{{\sin t}}{t}{\bf{j}}} \right) \cr
& {\text{Recall that }}\mathop {\lim }\limits_{t \to 0} \frac{{\sin t}}{t} = 1,{\text{ then}} \cr
& {\text{ }} = \sqrt 0 {\bf{i}} + 1{\bf{j}} \cr
& {\text{ }} = 0{\bf{i}} + {\bf{j}} \cr} $$
