limit of function at $x \rightarrow 2$

Question

ok, so this is a very basic question,

i'm trying to find the limit of the following function at $x \rightarrow 2$:

$|x^2 + 3x + 2| / (x^2 - 4)$

what i had previously done was simply plug in 2 for the LHL and the RHL, and write them off as negative infinite and positive infinite, respectively.

now im wondering, is there a smarter way to simplify this/get to the answer? i can break everything down to get $(x + 2)(x + 1) / (x + 2)(x - 2)$.

where would the negative sign from the absolute value come in? can i straight up cancel out the $(x + 2)$ terms from the numerator and denominator?

basically, my question is how do i get to the answer algebraically (which i'm pretty sure is what i am supposed to do)?

Answer 1

The limit is:

$$\lim_{x\to2}\frac{\left|x^2+3x+2\right|}{x^2-4}$$

Since the denominator $=0$ when $x=2$, the limit doesn't exist.