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For a fraction of polynomials: cancel common factors, then vertical asymptotes are where the bottom equals 0. The horizontal asymptote depends on degrees: if the bottom’s degree is bigger, \(y = 0\); if they are equal, \(y\) = ratio of leading coefficients. If the top’s degree is exactly one more, there is a slant asymptote instead.
How to Use the Asymptote Calculator
- Type a function, such as
(2x^2+1)/(x^2-4). - Press Solve.
- Read the vertical, horizontal, and slant asymptotes, plus any holes. Open the Graph tab to see the asymptotes as dashed lines.
What Is an Asymptote?
An asymptote is a line that a graph gets closer and closer to. Asymptotes come from limits:
- Vertical asymptote \(x = a\): the function shoots to \(\pm\infty\) as \(x \to a\).
- Horizontal asymptote \(y = L\): \(\lim_{x\to\pm\infty} f(x) = L\).
- Slant (oblique) asymptote \(y = mx + b\): the graph approaches a tilted line as \(x \to \pm\infty\).
Check any of these limits yourself with the limit calculator.
Rules for Rational Functions
| Compare degrees (top \(n\), bottom \(m\)) | Horizontal or slant asymptote |
|---|---|
| \(n < m\) | \(y = 0\) |
| \(n = m\) | \(y = \frac{\text{leading coefficient of top}}{\text{leading coefficient of bottom}}\) |
| \(n = m + 1\) | Slant asymptote: divide, and use the quotient |
| \(n > m + 1\) | None |
Worked Examples
Example 1: \(f(x) = \frac{2x^2 + 1}{x^2 – 4}\)
The bottom is zero at \(x = 2\) and \(x = -2\), and the top is not zero there. So the vertical asymptotes are \(x = -2\) and \(x = 2\). The degrees are equal, so the horizontal asymptote is \(y = \frac{2}{1} = 2\).
Example 2: \(f(x) = \frac{x^2 + 1}{x – 1}\)
Vertical asymptote: \(x = 1\). The top’s degree is one more than the bottom’s, so divide: \(x^2 + 1 = (x – 1)(x + 1) + 2\). The slant asymptote is \(y = x + 1\).
Example 3: \(f(x) = \frac{x^2 – 1}{x – 1}\)
Cancel first: \(\frac{(x-1)(x+1)}{x-1} = x + 1\). There is no vertical asymptote. Instead, the graph is the line \(y = x + 1\) with a hole at \(x = 1\).
Example 4: \(f(x) = \frac{e^x}{x – 1}\)
This is not a fraction of polynomials, so we use limits. As \(x \to -\infty\), \(e^x \to 0\), so \(y = 0\) is a horizontal asymptote on the left. There is a vertical asymptote at \(x = 1\).
Common Mistakes
- Calling a hole an asymptote. Always cancel common factors first.
- Thinking a graph can never cross a horizontal asymptote. It can, in the middle. The asymptote only describes the far ends.
- Looking for a horizontal and a slant asymptote at the same end. A graph has at most one of them on each side.
Learn the Math Behind the Asymptote Calculator
Asymptotes are found with limits: vertical ones where the function blows up, horizontal or slant ones as x grows very large.
- GuideHow to study calculus: a roadmap with checkpoints
- Key termsAsymptote, Limit
- Next toolsLimit Calculator, Simplify Calculator, Factoring Calculator
Frequently Asked Questions
How many vertical asymptotes can a function have?
Any number. \(\tan x\) has infinitely many. A fraction of polynomials has at most as many as the degree of its denominator.
What is the difference between a hole and a vertical asymptote?
At a hole, the function has a finite limit but is undefined at one point. At a vertical asymptote, the function grows without bound.
Can exponential or log functions have asymptotes?
Yes. \(e^x\) has the horizontal asymptote \(y = 0\), and \(\ln x\) has the vertical asymptote \(x = 0\).