AHL 2.13 (HL)—Further rational functions

Read asymptotes from a rational function’s leading structure.

Vertical asymptotes occur at non-cancelled denominator zeros; horizontal or oblique behaviour comes from comparing numerator and denominator degrees after simplification.

Worked example

For f(x)=(x²+1)/(x−2), x=2 is a vertical asymptote. Polynomial division gives f(x)=x+2+5/(x−2), so y=x+2 is the oblique asymptote.

A cancelled factor creates a hole, not a vertical asymptote; always simplify and record excluded domain values.

An asymptote describes limiting behaviour, not a value the function reaches at the asymptote.

Complete the graph-feature check for $f(x)=\frac{x^2+1}{x-2}=x+2+\frac5{x-2}$. Besides vertical asymptote $x=2$ and oblique asymptote $y=x+2$, the y-intercept is $f(0)=-1/2$. There are no real x-intercepts because $x^2+1=0$ has no real solution. Record all intercepts, asymptotes, holes and excluded inputs before sketching.