How To Identify the 3 Types of Non-Vertical Asymptotes

Non-Vertical (Horizontal and Slant/Oblique Asymptotes) are all about recognizing if a function is TOP-HEAVY, BOTTOM-HEAVY, OR BALANCED based on the degrees of x.

What I mean by “top-heavy” is that there is a higher degree of x in the numerator than in the denominator.

1. Top-Heavy

• What does it mean?
• there is a higher degree of x in the numerator than in the denominator
• ex: (x-5)³/(x²-4)
• What type of Non-Vertical Asymptote does it make?
• SLANT ASYMPTOTE (as long as the numerator is only 1 degree higher than the denominator)
• How do you find the equation?
• Do Long Division of the top divided by the bottom
• Ignore the remainder — this is not part of the equation

2. Bottom-Heavy

• What does it mean?
• there is a lower degree of x in the numerator than in the denominator
• ex: (x-5)²/(x³-4)
• What type of Non-Vertical Asymptote does it make?
• HORIZONTAL ASYMPTOTE
• How do you find the equation?
• The equation is always y=0 because you’re dividing by infinitely bigger and bigger numbersthat make the y value closer and closer to 0

3. Balanced

• What does it mean?
• there is an equal degree of x in the numerator than in the denominator
• ex: (x-5)²/(x²-4)
• What type of Non-Vertical Asymptote does it make?
• HORIZONTAL ASYMPTOTE
• How do you find the equation?
• The equation is going to be a ratio of the coefficients in front of the largest degrees of x
• ex: (3x³ — 4x² + x — 1) / (-2x³+8) would have a horizontal asymptote at y = -3/2