Answer
Prove that $\lim\limits_{t\to2}g(t)=g(2)$
Work Step by Step
*NOTES TO REMEMBER: $f(x)$ is continuous at $a$ if and only if $$\lim\limits_{x\to a}f(x)=f(a)$$
We consider
$\lim\limits_{t\to2}g(t)$
$=\lim\limits_{t\to2}\frac{t^2+5t}{2t+1}$
$=\frac{\lim\limits_{t\to2}(t^2+5t)}{\lim\limits_{t\to2}(2t+1)}$
$=\frac{\lim\limits_{t\to2}t^2+5\lim\limits_{t\to2}t}{2\lim\limits_{t\to2}t+\lim\limits_{t\to2}1}$
$=\frac{2^2+5\times2}{2\times2+1}$
$=g(2)$
Therefore, $g(t)$ is continuous at $2$.
