Find the limit involving absolute value and a squared term

Expert intro: This one is all about sign. The absolute value changes the algebra depending on which side of the point you approach from, so the shortcut is to split the expression first.

Detailed walkthrough

We want

$\lim_{x\to -2}\frac{(x+2)^2}{|x+2|}.$

Step 1: Simplify the quotient

For $x\neq -2$,

$\frac{(x+2)^2}{|x+2|}=|x+2|.$

Why? Because $(x+2)^2=(|x+2|)^2$.

Step 2: Take the limit

So the problem becomes

$\lim_{x\to -2}|x+2|.$

As $x\to -2$, we have $x+2\to 0$, hence

$|x+2|\to 0.$

Step 3: Final result

$\boxed{0}.$

💡 Pitfall guide

A classic mistake is splitting into cases too late and overcomplicating the algebra. Another is saying the limit does not exist because of the absolute value. Here the absolute value is harmless: after simplification, both sides go to 0.

🔄 Real-world variant

If the numerator were just $x+2$ instead of $(x+2)^2$, then

$\frac{x+2}{|x+2|}$

would approach $1$ from the right and $-1$ from the left, so the limit would fail to exist. The extra square is what removes the sign problem.

🔍 Related terms

absolute value, two-sided limit, sign analysis