9. Oblique Asymptote Finder — Theory
An oblique (slant) asymptote is a line that a function approaches as \(x\to \pm\infty\).
For rational functions, the fastest way to find it is polynomial division.
1) Division identity
\[
\begin{aligned}
\frac{P(x)}{D(x)} &= Q(x) + \frac{R(x)}{D(x)} \\
P(x) &= D(x)\cdot Q(x) + R(x) \\
\deg(R) &< \deg(D)
\end{aligned}
\]
The polynomial \(Q(x)\) is the polynomial asymptote. In the common “slant” case,
\(\deg(P)=\deg(D)+1\), so \(Q(x)\) is linear and the asymptote is a line \(y=mx+b\).
2) Why \(y=Q(x)\) is the asymptote
\[
\begin{aligned}
f(x) - Q(x) &= \frac{R(x)}{D(x)} \\
\deg(R) &< \deg(D) \;\Longrightarrow\; \frac{R(x)}{D(x)} \to 0 \quad (x\to \pm\infty) \\
\Rightarrow\quad f(x) &\to Q(x)
\end{aligned}
\]
3) Vertical asymptotes vs. holes
Domain breaks happen where \(D(x)=0\).
\[
\begin{aligned}
D(a)=0,\; P(a)\neq 0 &\Rightarrow \text{vertical asymptote at } x=a \\
D(a)=0,\; P(a)=0 &\Rightarrow \text{removable discontinuity (a hole) at } x=a
\end{aligned}
\]
4) Graph option: show \(P(x)\) and \(D(x)\)
If you enable the overlay option in the calculator, the graph can also plot the two polynomials \(y=P(x)\) and \(y=D(x)\).
This makes it visually clear that:
- Zeros of \(D(x)\) (where \(D(x)=0\)) are exactly the \(x\)-locations of vertical asymptotes or holes.
- The slant/polynomial asymptote comes from the quotient \(Q(x)\) produced by division.
Note: the overlay curves can grow very fast (especially higher degree), so the calculator provides an option to keep autoscaling focused
on the rational function and its asymptote unless you explicitly include \(P,D\) in autoscale.
