Horizontal and Vertical Asymptote Finder

Math Algebra • Polynomial and Rational Functions

Written by STEM Calculators Team Published December 14, 2025 Updated February 24, 2026

Enter polynomials in \(x\). Use ^ for powers and * for multiplication (implicit like 2x is accepted). Press Calculate to reduce \(\frac{P}{Q}\), detect vertical asymptotes (zeros of reduced \(Q\)), and compute horizontal/slant/polynomial asymptotes by degree comparison / division.

Numerator \(P(x)\): Denominator \(Q(x)\):

Graph window

\(x_{\min}\): \(x_{\max}\): Resolution (scan points): Root tolerance: Options:

Show steps Show slant/polynomial asymptote (if any) Auto-fit y-range Show probe point + coordinates Show holes (after cancellation)

Probe (evaluate at \(x_0\))

\(x_0\):

x0 = 0

Ready

Enter \(P(x)\) and \(Q(x)\), then press Calculate. The tool will:

Reduce \(\dfrac{P(x)}{Q(x)}\) by cancelling a polynomial GCD.
Report holes (cancelled roots) and vertical asymptotes (zeros of reduced \(Q\)).
Determine horizontal asymptote by degree comparison; show slant/polynomial asymptote if enabled.
Draw the graph with dashed asymptote lines and optional hole markers.

Drag to pan. Mouse wheel zooms (cursor-centered). Shift = y-zoom, Ctrl = gentler zoom. Double-click resets view. Dashed vertical lines are vertical asymptotes; dashed horizontal line is the horizontal asymptote; dashed slant/polynomial line appears if enabled.

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5. Horizontal And Vertical Asymptote Finder

A rational function is \[ f(x)=\frac{P(x)}{Q(x)} \] where \(P,Q\) are polynomials and \(Q(x)\neq 0\). Asymptotes describe the “end behavior” (horizontal/slant) or the “blow-up behavior” near zeros of the denominator (vertical).

1) First reduce: holes vs. vertical asymptotes

\[ \gcd(P,Q)=G,\qquad \frac{P}{Q}=\frac{P_{\text{red}}}{Q_{\text{red}}} \]

Holes (removable discontinuities): happen at zeros of \(G\). The reduced function gives the “missing” \(y\)-value.
Vertical asymptotes: happen at real zeros of \(Q_{\text{red}}(x)\).

2) Vertical asymptotes

If \(Q_{\text{red}}(a)=0\), then \(x=a\) is a vertical asymptote. The function typically diverges: \(\lim_{x\to a^-}f(x)=\pm\infty\) and/or \(\lim_{x\to a^+}f(x)=\pm\infty\). Multiplicity of the factor \((x-a)^m\) affects whether the graph crosses or “bounces,” but the asymptote still occurs.

3) Horizontal asymptotes (degree comparison)

\[ \deg P_{\text{red}} = m,\quad \deg Q_{\text{red}} = n \]

If \(m
If \(m=n\), then \(\displaystyle \lim_{x\to\pm\infty}f(x)=\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}\).
If \(m>n\), there is no horizontal asymptote (see slant/polynomial below).

4) Slant / polynomial asymptotes (long division)

When \(m\ge n\), divide:

\[ \frac{P_{\text{red}}(x)}{Q_{\text{red}}(x)} = S(x) + \frac{R(x)}{Q_{\text{red}}(x)} \]

If \(\deg R < \deg Q_{\text{red}}\), then \(S(x)\) is the asymptote:

If \(\deg S = 1\), it’s a slant asymptote.
If \(\deg S \ge 2\), it’s a polynomial asymptote.

5) What the calculator plots

The reduced function \(P_{\text{red}}/Q_{\text{red}}\) as a continuous curve (breaking near vertical asymptotes).
Dashed lines: vertical asymptotes \(x=a\), horizontal \(y=c\), and optional \(y=S(x)\).
Optional hole markers: open circles at cancelled roots with coordinates from the reduced function.

Tip: if you want to explore behavior “toward infinity,” widen the graph window (or use the tool’s Zoom to infinity button).

Frequently Asked Questions

How does the calculator find vertical asymptotes of a rational function?

It first cancels any common polynomial factors of P(x) and Q(x). Vertical asymptotes occur at real x-values where the reduced denominator Q_red(x) = 0.

What is the difference between a hole and a vertical asymptote?

A hole (removable discontinuity) happens when a factor cancels between numerator and denominator, so the original function is undefined at that x even though the reduced function has a finite value. A vertical asymptote happens when the reduced denominator is zero, causing the function to diverge near that x-value.

How is the horizontal asymptote determined for P(x)/Q(x)?

Let m = deg(P_red) and n = deg(Q_red). If m < n, the horizontal asymptote is y = 0; if m = n, the horizontal asymptote is y = (leading coefficient of P_red) / (leading coefficient of Q_red); if m > n, there is no horizontal asymptote.

When does a slant or polynomial asymptote appear instead of a horizontal asymptote?

When the numerator degree is at least the denominator degree, the calculator can use long division to write P_red/Q_red = S(x) + R(x)/Q_red(x). If deg(R) < deg(Q_red), then y = S(x) is the slant asymptote (deg(S)=1) or a polynomial asymptote (deg(S) >= 2).

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Frequently Asked Questions

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