Answer
$$ - {\bf{i}} - 4{\bf{j}} + {\bf{k}}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{t \to \pi /2} \left( {\cos 2t\,{\bf{i}} - 4\sin t\,{\bf{j}} + \frac{{2t}}{\pi }\,{\bf{k}}} \right) \cr
& {\text{use the sum limits property }}\left( {{\text{see page 805}}} \right) \cr
& = \mathop {\lim }\limits_{t \to \pi /2} \left( {\cos 2t\,} \right){\bf{i}} - 4\mathop {\lim }\limits_{t \to \pi /2} \left( {\sin t\,} \right){\bf{j}} + \mathop {\lim }\limits_{t \to \pi /2} \left( {\frac{{2t}}{\pi }\,} \right){\bf{k}} \cr
& {\text{evalute each limit substitute }}\pi {\text{/2 for }}t \cr
& = \left( {\cos 2\left( {\frac{\pi }{2}} \right)\,} \right){\bf{i}} - 4\left( {\sin \frac{\pi }{2}} \right){\bf{j}} + \left( {\frac{2}{\pi }\left( {\frac{\pi }{2}} \right)\,} \right){\bf{k}} \cr
& {\text{simplifying}} \cr
& = \left( { - 1} \right){\bf{i}} - 4\left( 1 \right){\bf{j}} + \left( 1 \right){\bf{k}} \cr
& - {\bf{i}} - 4{\bf{j}} + {\bf{k}} \cr} $$
