Survey y = (x² + x + 1)/(x − 1)
Equation: y = (x² + x + 1)/(x − 1)
Polynomial division gives y = x + 2 + 3/(x − 1). The oblique asymptote is y = x + 2; the vertical one is x = 1.
The derivative changes sign on each side of the asymptote, so each branch has one turning point.