Answer
$$\left\langle {\frac{1}{3},0} \right\rangle $$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{t \to + \infty } \left\langle {\frac{{{t^2} + 1}}{{3{t^2} + 2}},\frac{1}{t}} \right\rangle \cr
& {\text{Evaluate the limit}} \cr
& \mathop {\lim }\limits_{t \to + \infty } \left\langle {\frac{{{t^2} + 1}}{{3{t^2} + 2}},\frac{1}{t}} \right\rangle = \left\langle {\mathop {\lim }\limits_{t \to + \infty } \frac{{{t^2} + 1}}{{3{t^2} + 2}},\mathop {\lim }\limits_{t \to + \infty } \frac{1}{t}} \right\rangle \cr
& {\text{ }} = \left\langle {\frac{{{{\left( { + \infty } \right)}^2} + 1}}{{3{{\left( \infty \right)}^2} + 2}},\frac{1}{{ + \infty }}} \right\rangle \cr
& {\text{Simplifying}} \cr
& {\text{ }} = \left\langle {\frac{\infty }{\infty },0} \right\rangle \cr
& {\text{Therefore,}} \cr
& \mathop {\lim }\limits_{t \to + \infty } \left\langle {\frac{{{t^2} + 1}}{{3{t^2} + 2}},\frac{1}{t}} \right\rangle = \mathop {\lim }\limits_{t \to + \infty } \left\langle {\frac{{\frac{{{t^2}}}{{{t^2}}} + \frac{1}{{{t^2}}}}}{{\frac{{3{t^2}}}{{{t^2}}} + \frac{2}{{{t^2}}}}},\frac{1}{t}} \right\rangle \cr
& {\text{ }} = \mathop {\lim }\limits_{t \to + \infty } \left\langle {\frac{{1 + \frac{1}{{{t^2}}}}}{{3 + \frac{2}{{{t^2}}}}},\frac{1}{t}} \right\rangle \cr
& {\text{ }} = \left\langle {\frac{{1 + \frac{1}{{{{\left( \infty \right)}^2}}}}}{{3 + \frac{2}{{{{\left( \infty \right)}^2}}}}},\frac{1}{\infty }} \right\rangle \cr
& {\text{ }} = \left\langle {\frac{1}{3},0} \right\rangle \cr} $$
