4. Asymptote And Hole Detector — Theory
This tool analyzes rational functions \(f(x)=\dfrac{P(x)}{Q(x)}\) to find holes and asymptotes
using cancellation, degree comparison, and polynomial long division.
0) What this calculator supports
The detector is designed for rational functions (quotients of polynomials):
\[
f(x)=\frac{P(x)}{Q(x)},\qquad P,Q\text{ polynomials, }Q(x)\neq 0.
\]
If your expression contains non-polynomial pieces (like \(\sqrt{x}\), \(\ln x\), \(\sin x\), etc.), the standard “rational” rules below may not apply.
1) Holes (removable discontinuities)
A hole occurs when numerator and denominator share a common factor \((x-a)\) that cancels:
\[
f(x)=\frac{(x-a)\,R(x)}{(x-a)\,S(x)}=\frac{R(x)}{S(x)} \quad (x\ne a)
\]
The original function is not defined at \(x=a\), but the limit exists and equals the simplified value:
\[
\lim_{x\to a} f(x) = \frac{R(a)}{S(a)}.
\]
On the graph, a hole is shown as an open circle at
\(\bigl(a,\frac{R(a)}{S(a)}\bigr)\).
To “fix” the discontinuity, you can redefine the function by setting \(f(a)=\frac{R(a)}{S(a)}\).
Example
\[
\frac{x^2-4}{x-2}=\frac{(x-2)(x+2)}{x-2}=x+2 \quad (x\ne 2)
\]
Hole at \(x=2\), with coordinate \((2,4)\). After cancellation the graph is the line \(y=x+2\) with a missing point.
2) Vertical asymptotes
A vertical asymptote happens where the reduced denominator is zero and the factor does not cancel.
If the simplified function is \(f_{\text{simp}}(x)=\dfrac{P_{\text{red}}(x)}{Q_{\text{red}}(x)}\), then:
\[
Q_{\text{red}}(a)=0 \quad \Longrightarrow \quad x=a \text{ is a vertical asymptote.}
\]
Intuition: near \(x=a\), the denominator gets very small while the numerator stays finite, so \(f(x)\) grows without bound.
On the graph, vertical asymptotes are drawn as dashed vertical lines.
Example
\[
f(x)=\frac{1}{x-3} \quad \Rightarrow \quad x=3 \text{ is a vertical asymptote.}
\]
3) Horizontal asymptotes (degree comparison)
For a reduced rational function \(f(x)=\dfrac{P(x)}{Q(x)}\), compare degrees:
\(\deg P=n\), \(\deg Q=m\).
\[
\deg P < \deg Q \Rightarrow \lim_{x\to\pm\infty} \frac{P(x)}{Q(x)} = 0 \quad \Rightarrow \quad \text{horizontal asymptote } y=0
\]
\[
\deg P = \deg Q \Rightarrow \lim_{x\to\pm\infty} \frac{P(x)}{Q(x)} =
\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}
\quad \Rightarrow \quad \text{horizontal asymptote } y=\frac{a_n}{b_m}
\]
Horizontal asymptotes are drawn as dashed horizontal lines.
Examples
\[
\frac{2x^2+1}{x^2-3} \to \frac{2}{1}=2 \quad\Rightarrow\quad y=2
\]
\[
\frac{x+1}{x^3+5} \to 0 \quad\Rightarrow\quad y=0
\]
4) Slant and polynomial asymptotes (long division)
If \(\deg P \ge \deg Q\), use polynomial long division:
\[
\frac{P(x)}{Q(x)} = q(x) + \frac{r(x)}{Q(x)},\qquad \deg r < \deg Q.
\]
As \(x\to\pm\infty\), the remainder term \(\dfrac{r(x)}{Q(x)}\to 0\), so the asymptote is:
\[
y = q(x).
\]
-
If \(\deg P = \deg Q + 1\), then \(q(x)\) is linear, so \(y=mx+b\) is a slant (oblique) asymptote.
-
If \(\deg P \ge \deg Q + 2\), then \(q(x)\) has degree \(\ge 2\), so \(y=q(x)\) is a polynomial (curved) asymptote.
Examples
\[
\frac{x^2+1}{x-1} = x+1 + \frac{2}{x-1}
\quad\Rightarrow\quad \text{slant asymptote } y=x+1
\]
\[
\frac{x^4+1}{x^2+1} = x^2 - 1 + \frac{2}{x^2+1}
\quad\Rightarrow\quad \text{polynomial asymptote } y=x^2-1
\]
5) Special case: the function simplifies to a polynomial
If after cancellation the denominator becomes a nonzero constant (or the division remainder is exactly zero),
then \(f(x)\) is exactly a polynomial.
Polynomials have no vertical, horizontal, or slant asymptotes.
You may still have a hole if cancellation removed a factor from the original denominator.
