A missing point and a vertical asymptote are different kinds of failure.
WHAT HAPPENS IF…At a=-2 and b=1, what is the height of the hole?
Interactive exploration
Loading the mathematical model…
Connect the picture to the ideaf(x)=(x−a)(x−b)(x−a)(x+1),x=a,b
Away from the excluded inputs, cancellation gives (x+1)/(x−b). At x=a this simplified expression has a finite limit, so the original graph has a hole. Since b>0 here, its numerator at b is nonzero, producing a pole.
A detail that matters
Cancellation changes the expression, not the original domain.