Vertical asymptote
Find asymptotes of 1/(x-2)
Method: Set the uncanceled denominator equal to zero.
- 1The denominator vanishes at x=2.
- 2No numerator factor cancels x-2.
- 3The function grows without bound near x=2.
Final answer
Free online calculator
Asymptote Calculator
Analyze rational functions to find vertical, horizontal, and slant asymptotes while excluding removable holes.
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How to Use the Asymptote Calculator
1
Enter a rational function
Use a numerator divided by a denominator in x.
2
Analyze the denominator
Real denominator zeros are checked for vertical asymptotes or holes.
3
Compare polynomial degrees
Degree and division rules determine horizontal or slant behavior.
Example
How the Asymptote Calculator Chooses a Method
Find vertical, horizontal, or slant asymptotes and distinguish them from removable holes.
The calculator factors the expression, checks denominator zeros that remain after cancellation, and compares polynomial degrees for end behavior. A canceled denominator factor identifies a hole rather than a vertical asymptote.
Supported problem types
- Vertical asymptotes of rational functions
- Horizontal asymptotes from end behavior
- Slant asymptotes from polynomial division
Limits and assumptions
- A denominator zero is not automatically an asymptote if its factor cancels.
- Non-rational functions may need separate limit analysis.
Worked Asymptote Calculator Examples
Each example names the method, shows the ordered reasoning, and keeps the final answer separate so you can check your own work.
Horizontal asymptote
Find the horizontal asymptote of (2x+1)/(x-3)
Method: Compare equal polynomial degrees.
- 1Both numerator and denominator have degree 1.
- 2Use the ratio of leading coefficients 2/1.
- 3The end behavior approaches y=2.
Final answer
Hole versus asymptote
Analyze (x^2-1)/(x-1)
Method: Factor and cancel before classifying denominator zeros.
- 1Factor the numerator as (x-1)(x+1).
- 2Cancel x-1 for x not equal to 1.
- 3The missing point is a hole at (1,2), not a vertical asymptote.
Final answer
Learn This Topic
What the Asymptote Calculator Helps You Understand
Analyze rational functions to find vertical, horizontal, and slant asymptotes while excluding removable holes.
Learning focus 1
Use a numerator divided by a denominator in x.
Learning focus 2
Real denominator zeros are checked for vertical asymptotes or holes.
Learning focus 3
Degree and division rules determine horizontal or slant behavior.
Input tips
- • Use a numerator divided by a denominator in x.
- • It is a vertical line x = a approached by the function when the denominator tends to zero without cancellation.
Common mistakes to avoid
- • Treating a removable hole as a vertical asymptote.
- • Using only leading coefficients without first comparing polynomial degrees.
Frequently Asked Questions
What is a vertical asymptote?
It is a vertical line x = a approached by the function when the denominator tends to zero without cancellation.
Can a function have a slant asymptote?
Yes. A slant asymptote occurs when the numerator degree is exactly one greater than the denominator degree.
Is every denominator zero a vertical asymptote?
No. First factor and simplify; a canceled factor creates a removable hole instead.
Can a graph cross a horizontal asymptote?
Yes. A horizontal asymptote describes end behavior and does not prevent crossings at finite x-values.