Answer
$g$ is continuous at $a=2$
Work Step by Step
A function $f$ is continuous at a number $a$ if
$\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$
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$\displaystyle \lim_{t\rightarrow 2}g(t)=\lim_{t\rightarrow 2}\frac{t^{2}+5t}{2t+1}=\quad ...$The limit of a quotient
$=\displaystyle \frac{\lim_{t\rightarrow 2}(t^{2}+5t)}{\lim_{t\rightarrow 2}(2t+1)}= \quad $...The limit of a sum (twice)
$=\displaystyle \frac{\lim_{t\rightarrow 2}t^{2}+\lim_{t\rightarrow 2}5t}{\lim_{t\rightarrow 2}2t+\lim_{t\rightarrow 2}1}\quad $...The limit of a constant times a function
$=\displaystyle \frac{\lim_{t\rightarrow 2}t^{2}+5\lim_{t\rightarrow 2}t}{2\lim_{t\rightarrow 2}t+\lim_{t\rightarrow 2}1}\quad $...evaluate
$=\displaystyle \frac{2^{2}+5(2)}{2(2)+1}$
$=\displaystyle \frac{14}{5}$
$g(2)=\displaystyle \frac{2^{2}+5(2)}{2(2)+1}=\frac{14}{5}=\lim_{t\rightarrow 2}g(t).$
By the definition, $g$ is continuous at $a=2$.
