Question

How do I find these limits and their horizontal asymptotes without graphing them?

Answer 1

\[\lim_{x\to\infty}\frac{3x+1}{|x|+2}\]
Since \(x\to\infty\), you're dealing with strictly positive numbers, which means \(|x|=x\).
So you have
\[\lim_{x\to\infty}\frac{3x+1}{x+2}\]

Similarly (but different), the second limit has \(x\to-\infty\), which means \(x\) is strictly negative, so \(|x|=-x\). And you have the limit
\[\lim_{x\to-\infty}\frac{3x+1}{-x+2}\]

Answer 2

Awesome. That made it a lot simpler. Thanks so much! Would that work for this one too?

Answer 3

Yep, \(x\to\infty~~\Rightarrow~~x>0~~\Rightarrow~~|x|=x\).