Answer
$-\dfrac{1}{4}$
Work Step by Step
We will use the following limit rules for this problem:
$\ Rule \ - \ 1 : \lim\limits_{x \to a} A(x)=A(a)$
$\ Rule \ - \ 2: \lim\limits_{x \to a} \dfrac{A(x)}{B(x)}=\dfrac{\lim\limits_{x \to a} A(x)}{\lim\limits_{x \to a} B(x)}$
Apply the given rules:
$\lim\limits_{x\to -1}\dfrac{x^2+x+2}{x^2-9}=\dfrac{\lim\limits_{x\to -1} (x^2+x+2)}{\lim\limits_{x\to -1}(x^2-9) } \\=\dfrac{(-1)^2+(-1)+2}{(-1)^2-9}\\=-\dfrac{1}{4}$
