Let f(x) = (x² - x - 2)/(x² - 4) wherever it is defined…

Step 1. Factorise. $f(x) = \dfrac{(x - 2)(x + 1)}{(x - 2)(x + 2)}$. It is undefined at $x = 2$ and at $x = -2$.

Step 2. The point $x = 2$. For $x \ne 2$, $f(x) = \dfrac{x + 1}{x + 2}$. As $x \to 2$ this tends to $\dfrac{3}{4} = 0.75$. The limit exists, so this break is removable.

Step 3. The point $x = -2$. Near $-2$ the simplified form $\dfrac{x + 1}{x + 2}$ has top $-1$ and a bottom that tends to zero. The values grow without bound, so this break is infinite.