Example Find the limit of the following function．．

$$\displaystyle{\lim_{x \to 2} \frac{x^2 - 3x + 2}{x^2 - 4}}$$

Answer To find the limit of a function, it is imporatnt to know the followings:

1. The limit of a sum is the sum of the limits． $\lim_{x \to a}(f(x) + g(x)) = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)$
2. The limit of a constant multiple is the constant times the limit． $\lim_{x \to a}(\alpha f(x)) = \alpha \lim_{x \to a} f(x)$
3. $\lim_{x \to a}f(x)g(x) = \lim_{x \to a}f(x) \lim_{x \to a}g(x)$
4. $\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}$. Here, $\lim_{x \to a}g(x) \neq 0$
5. Basic formula for limit. $\lim_{x \to 0}\frac{\sin{x}}{x} = 1$

Apply these rules to the above problem, we have

$$\lim_{x \to 2}\frac{x^2 - 3x + 2}{x^2 - 4} = \lim_{x \to 2}\frac{(x-1)(x-2)}{(x+2)(x-2)} = \lim_{x \to 2}\frac{x-1}{x + 2} = \frac{1}{4}.$$