Asymptote Calculator
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An asymptote is a line the graph approaches. For \(f(x) = \frac{x^{2} – 1}{x – 2}\): the denominator is 0 at x = 2, so x = 2 is a vertical asymptote, and dividing gives \(x + 2 + \frac{3}{x – 2}\), so y = x + 2 is a slant asymptote.
\(f(x) = \frac{x^{2} – 1}{x – 2}\)Rational functions get exact answers. Others (e^x, ln x, tan x) are handled with limits.
Graph of f (green) with vertical asymptotes (red), horizontal (blue) and slant (purple) asymptotes dashed.
| Vertical asymptotes | \(x = 2\) |
| Horizontal asymptote | none |
| Slant asymptote | \(y = x + 2\) |
| x-intercepts | \((-1,\ 0)\), \((1,\ 0)\) |
| y-intercept | \((0,\ \frac{1}{2})\) |
| Domain | \(x \neq 2\) |
Show steps
- Write f(x) as one fraction: \(f(x) = \frac{x^{2} – 1}{x – 2}\)
- Vertical asymptotes are where the simplified denominator is zero: \(x – 2 = 0 \Rightarrow x = 2\)
- The numerator’s degree is one more than the denominator’s, so there is a slant (oblique) asymptote. Divide: \(\frac{x^{2} – 1}{x – 2} = x + 2 + \frac{3}{x – 2}\)
- As x → ±∞ the remainder fraction goes to 0, so the slant asymptote is \(y = x + 2\).
How to use the asymptote calculator
- Enter f(x), for example (2x^2 + 3)/(x^2 − 4).
- Press Find asymptotes. You get the vertical, horizontal and slant asymptotes, any holes, the intercepts and the domain.
- The graph shows vertical asymptotes in red, horizontal in blue and slant in purple.
Vertical asymptotes and holes
Factor and cancel common factors first. Values that make the simplified denominator 0 are vertical asymptotes. Values that made the original denominator 0 but cancelled are holes.
\(\frac{x+3}{x^{2}+5x+6} = \frac{x+3}{(x+2)(x+3)} = \frac{1}{x+2}\): vertical asymptote x = −2, hole at \(\left(-3, -1\right)\).
Horizontal and slant asymptotes
Compare the degree n of the numerator with the degree d of the denominator:
| Degrees | Asymptote | Example |
|---|---|---|
| n < d | y = 0 | \(\frac{3x}{x^{2}+1}\) |
| n = d | y = ratio of leading coefficients | \(\frac{2x^{2}+3}{x^{2}-4} \Rightarrow y = 2\) |
| n = d + 1 | slant: y = quotient of the division | \(\frac{x^{2}-1}{x-2} \Rightarrow y = x + 2\) |
| n > d + 1 | none (polynomial-like end behavior) | \(\frac{x^{3}}{x+1}\) |
Other functions
- \(e^{x}\) has the horizontal asymptote y = 0 as x → −∞; \(\frac{e^{x}}{e^{x}+1}\) has y = 0 and y = 1.
- \(\ln x\) has the vertical asymptote x = 0.
- \(\tan x\) has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\).
Common mistakes
- Calling a hole a vertical asymptote: always cancel common factors first.
- Thinking a graph can never cross a horizontal asymptote; it can, for finite x.
- Looking for both a horizontal and a slant asymptote on the same side for a rational function; there is at most one.
Asymptotes practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
The asymptote describes behavior as x → ±∞; crossings at finite x are allowed.
Quiz complete
Frequently asked questions
How do you find vertical asymptotes?
Simplify the function, then set the denominator equal to zero. Each real solution that does not cancel is a vertical asymptote.
How do you find the horizontal asymptote?
Compare degrees: if the numerator’s degree is smaller, y = 0; if they are equal, y is the ratio of leading coefficients; if the numerator’s is larger, there is no horizontal asymptote.
When is there a slant asymptote?
When the numerator’s degree is exactly one more than the denominator’s. Divide the polynomials; the quotient is the slant asymptote.
What is the difference between a hole and an asymptote?
A hole comes from a factor that cancels; the graph just has a missing point. An asymptote comes from a factor that stays in the denominator; the graph shoots off to infinity.
Does it work for non-rational functions?
Yes. It finds horizontal asymptotes from limits at ±∞ and vertical asymptotes where the function grows without bound, and reports repeating ones like tan x with a general formula.
Sources
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