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Free slant asymptote calculator & oblique asymptote calculator. Calculate slant asymptotes of rational functions using polynomial long division with step-by-step solutions. Our calculator finds diagonal asymptotes when the numerator degree is exactly one more than the denominator degree.
Last updated: February 2, 2026
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Use standard notation: x^2 for x², x^3 for x³, etc.
Use standard notation: x^2 for x², x^3 for x³, etc.
The function has a slant asymptote. Perform polynomial long division to find the exact equation.
Condition for slant asymptote
Degree difference = 1
Calculates slant asymptotes when numerator degree exceeds denominator degree by exactly 1
Asymptote type
Diagonal Line
Finds diagonal asymptotes that are neither horizontal nor vertical
Asymptote types
Horizontal, Vertical, Slant
Identifies and calculates all possible asymptote types for rational functions
Division process
Quotient + Remainder
Shows step-by-step polynomial division with quotient and remainder
Quick analysis
Degree Check → Division
Automatically determines if slant asymptote exists and calculates it
Comprehensive analysis
All Asymptotes + Behavior
Analyzes complete behavior including all asymptote types and end behavior
For function f(x) = (x² + 1)/(x + 1):
Degree difference
1
Slant asymptote
y = x - 1
Our slant asymptote calculator uses polynomial long division to find diagonal asymptotes of rational functions. The calculation applies degree analysis principles to determine when slant asymptotes exist and calculates their exact equations.
Degree difference = 1 → Slant asymptote existsDegree difference = 0 → Horizontal asymptoteDegree difference > 1 → No slant asymptoteDegree difference < 0 → Horizontal asymptote at y = 0This fundamental rule determines when slant asymptotes exist by comparing the degrees of the numerator and denominator polynomials in the rational function.
Shows how rational functions approach diagonal asymptotes at infinity
Slant asymptote analysis is based on polynomial long division and limit theory. When the numerator degree exceeds the denominator degree by exactly 1, the quotient of the division gives the slant asymptote equation. The remainder approaches zero as x approaches infinity, leaving only the linear asymptote.
Need help with other function analysis? Check out our end behavior calculator and concavity calculator.
Get Custom Calculator for Your PlatformResult: Slant asymptote is y = x - 1
As x → ±∞, f(x) approaches the line y = x - 1. The remainder 2/(x + 1) approaches 0.
f(x) = (x³ - 2x + 1)/(x² - 1)
Slant asymptote: y = x
f(x) = (x² + 1)/(x² - 1)
Horizontal asymptote: y = 1
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