Mathematics

The key is to check whether the function blows up to infinity as you approach $x=3$, or whether it actually settles to a finite value. For $x<3$, simplify the rational expression by factoring—you'll find the apparent discontinuity cancels out, giving you a real limit. Then compare that limit to what happens from the right at $x=3$. A vertical asymptote only exists if the function diverges to $\pm\infty$; if both one-sided limits are finite, there's no asymptote, just a removable discontinuity or jump.

The key is to check whether the function blows up to infinity as you approach $x=3$, or whether it actually settles to a finite value. For $x<3$, simplify the rational expression by factoring—you'll find the apparent discontinuity cancels out, giving you a real limit. Then compare that limit to what happens from the right at $x=3$. A vertical asymptote only exists if the function diverges to $\pm\infty$; if both one-sided limits are finite, there's no asymptote, just a removable discontinuity or jump.