Answer
$\frac{{37}}{4}$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{t \to 7} \frac{{3{t^2} + 1}}{{{t^2} - 5t + 2}} \cr
& {\text{Suppose that }}c{\text{ is a constant and the limits }} \cr
& {\text{ }}\mathop {\lim }\limits_{x \to a} f\left( x \right){\text{ and }}\mathop {\lim }\limits_{x \to a} g\left( x \right) \cr
& \cr
& {\text{Use the Quotient Law}} \cr
& \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}} = \frac{{\mathop {\lim }\limits_{x \to a} f\left( x \right)}}{{\mathop {\lim }\limits_{x \to a} g\left( x \right)}},{\text{ }}\mathop {\lim }\limits_{x \to a} g\left( x \right) \ne 0 \cr
& \mathop {\lim }\limits_{t \to 7} \frac{{3{t^2} + 1}}{{{t^2} - 5t + 2}} = \frac{{\mathop {\lim }\limits_{t \to 7} \left( {3{t^2} + 1} \right)}}{{\mathop {\lim }\limits_{t \to 7} \left( {{t^2} - 5t + 2} \right)}} \cr
& \cr
& {\text{Use the Sum and Difference Laws }}\left( {{\text{see page 95}}} \right) \cr
& \mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right) \pm g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) \pm \mathop {\lim }\limits_{x \to a} g\left( x \right) \cr
& \frac{{\mathop {\lim }\limits_{t \to 7} \left( {3{t^2} + 1} \right)}}{{\mathop {\lim }\limits_{t \to 7} \left( {{t^2} - 5t + 2} \right)}} = \frac{{\mathop {\lim }\limits_{t \to 7} \left( {3{t^2}} \right) + \mathop {\lim }\limits_{t \to 7} \left( 1 \right)}}{{\mathop {\lim }\limits_{t \to 7} \left( {{t^2}} \right) - \mathop {\lim }\limits_{t \to 7} \left( {5t} \right) + \mathop {\lim }\limits_{t \to 7} \left( 2 \right)}} \cr
& \cr
& {\text{Use the Constant Multiple Law }}\left( {{\text{see page 95}}} \right) \cr
& \mathop {\lim }\limits_{x \to a} \left[ {cf\left( x \right)} \right] = c\mathop {\lim }\limits_{x \to a} f\left( x \right) \cr
& = \frac{{\mathop {3\lim }\limits_{t \to 7} \left( {{t^2}} \right) + \mathop {\lim }\limits_{t \to 7} \left( 1 \right)}}{{\mathop {\lim }\limits_{t \to 7} \left( {{t^2}} \right) - 5\mathop {\lim }\limits_{t \to 7} \left( t \right) + \mathop {\lim }\limits_{t \to 7} \left( 2 \right)}} \cr
& \cr
& {\text{Use The power Law }}\mathop {\lim }\limits_{x \to a} {x^n} = {a^n}{\text{, }}n{\text{ is a positive integer }} \cr
& {\text{and the laws }}\mathop {\lim }\limits_{x \to a} x = a,{\text{ }}\mathop {\lim }\limits_{x \to a} c = c,{\text{ then}} \cr
& = \frac{{3{{\left( 7 \right)}^2} + \left( 1 \right)}}{{{{\left( 7 \right)}^2} - 5\left( 7 \right) + 2}} \cr
& {\text{Simplifying}} \cr
& = \frac{{148}}{{49 - 35 + 2}} \cr
& = \frac{{37}}{4} \cr} $$
