oblique asymptote calculator

What is an oblique asymptote?

An oblique asymptote (also called a slant asymptote) is a line that a rational function approaches as x goes to positive or negative infinity. It usually appears in the form y = mx + b, where m is the slope and b is the intercept.

For a rational function f(x) = P(x)/Q(x), an oblique asymptote exists when the degree of P(x) is exactly one greater than the degree of Q(x).

How this oblique asymptote calculator works

This calculator performs polynomial long division of the numerator by the denominator.

• If the degree difference is exactly 1, the quotient is a line, and that line is the oblique asymptote.
• If the degree difference is less than 1, there is no oblique asymptote.
• If the degree difference is greater than 1, the asymptote is polynomial (not oblique/slant).

You also get the quotient and remainder so you can see the full decomposition:

f(x) = quotient + remainder / denominator

Step-by-step example

Example function

Consider:

f(x) = (3x³ + 2x² - 5) / (x² - 1)

1) Compare degrees

Degree of numerator = 3, degree of denominator = 2. Difference = 1, so an oblique asymptote should exist.

2) Divide polynomials

Long division gives:

f(x) = 3x + 2 + (3x - 3)/(x² - 1)

3) Identify asymptote

As x becomes large, the fraction term approaches 0, so the function approaches:

y = 3x + 2

Common mistakes to avoid

• Entering coefficients in the wrong order (must be highest power to constant).
• Forgetting zero placeholders for missing powers (e.g., 2,0,-7 for 2x² - 7).
• Assuming every rational function has a slant asymptote.
• Confusing vertical asymptotes (zeros of denominator) with oblique asymptotes (end behavior).

Quick FAQ

Can a function have both vertical and oblique asymptotes?

Yes. Vertical asymptotes come from denominator zeros; oblique asymptotes come from degree behavior at infinity.

What if the quotient is quadratic?

Then the function has a polynomial asymptote, not an oblique (linear) asymptote.

Do I need to simplify first?

It is a good idea. If factors cancel, the function may have holes and different asymptotic behavior than expected.