Answer
Prove that $\lim\limits_{x\to a}g(x)=g(a)$ for $\forall a\in(-\infty,-2)$
Work Step by Step
*NOTES TO REMEMBER: $f(x)$ is continuous on the interval if and only if it is continuous at every point in the interval.
In other words, $f(x)$ is continuous on the interval $(u,v)$ if and only if for $\forall a\in(u,v)$, we have $$\lim\limits_{x\to a}f(x)=f(a)$$
For $\forall a\in(-\infty,-2)$, we consider
$\lim\limits_{x\to a}g(x)$
$=\lim\limits_{x\to a}\frac{x-1}{3x+6}$
$=\frac{\lim\limits_{x\to a}(x-1)}{\lim\limits_{x\to a}(3x+6)}$
$=\frac{\lim\limits_{x\to a}x-\lim\limits_{x\to a}1}{3\lim\limits_{x\to a}x+\lim\limits_{x\to a}6}$
$=\frac{a-1}{3a+6}$
$=g(a)$
Therefore, $g(x)$ is continuous on the interval $(-\infty,-2)$.
