Answer
$0{\bf{i}} + 0{\bf{j}} + 0{\bf{k}}$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{t \to \infty } \left( {{e^{ - t}}{\bf{i}} + \frac{1}{t}{\bf{j}} + \frac{t}{{{t^2} + 1}}{\bf{k}}} \right) \cr
& {\text{Distribute using the limit properties}} \cr
& = \mathop {\lim }\limits_{t \to \infty } \left( {{e^{ - t}}} \right){\bf{i}} + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{1}{t}} \right){\bf{j}} + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{t}{{{t^2} + 1}}} \right){\bf{k}} \cr
& = \mathop {\lim }\limits_{t \to \infty } \left( {{e^{ - t}}} \right){\bf{i}} + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{1}{t}} \right){\bf{j}} + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{{\frac{t}{{{t^2}}}}}{{\frac{{{t^2}}}{{{t^2}}} + \frac{1}{{{t^2}}}}}} \right){\bf{k}} \cr
& = \mathop {\lim }\limits_{t \to \infty } \left( {{e^{ - t}}} \right){\bf{i}} + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{1}{t}} \right){\bf{j}} + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{{\frac{1}{t}}}{{1 + \frac{1}{{{t^2}}}}}} \right){\bf{k}} \cr
& = \mathop {\lim }\limits_{t \to \infty } \left( {{e^{ - t}}} \right){\bf{i}} + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{1}{t}} \right){\bf{j}} + \left( {\frac{{\mathop {\lim }\limits_{t \to \infty } \frac{1}{t}}}{{\mathop {\lim }\limits_{t \to \infty } \left( 1 \right) + \mathop {\lim }\limits_{t \to \infty } \left( {\frac{1}{{{t^2}}}} \right)}}} \right){\bf{k}} \cr
& {\text{Evaluate the limit when }}t \to \infty \cr
& = 0{\bf{i}} + 0{\bf{j}} + \left( {\frac{0}{{1 + 0}}} \right){\bf{k}} \cr
& {\text{Simplifying}} \cr
& = 0{\bf{i}} + 0{\bf{j}} + 0{\bf{k}} \cr} $$
