Slant Asymptotes: How to Find Them

A slant asymptote (or oblique asymptote) is a slanted line $y = mx + b$ that the graph of a rational function approaches as $x$ goes to $\infty$ or $-\infty$ . It happens when the numerator’s degree is exactly one more than the denominator’s. To find it, divide the numerator by the denominator; the quotient, without the remainder, is the asymptote.

Before this lesson: You should know Graphing Rational Functions, Polynomial Long Division and Synthetic Division.

Why Some Graphs Have a Slant Asymptote

Division splits a rational function into a polynomial part plus a leftover fraction. For example, $\frac{x^2 + 3x + 5}{x + 1} = x + 2 + \frac{3}{x + 1}$ .

As $x$ gets very large, positive or negative, the leftover fraction $\frac{3}{x + 1}$ shrinks toward 0. So the graph gets closer and closer to the line $y = x + 2$ . That line is the slant asymptote.

If the top’s degree is one more than the bottom’s, the polynomial part is a line, so the asymptote is slanted. If the degrees are equal, the polynomial part is a constant, which gives a horizontal asymptote instead.

Graph of f(x) = (x² + 3x + 5)/(x + 1) in solid blue, in two branches. A dashed orange slant line y = x + 2 runs diagonally, and both branches get closer to it far to the left and far to the right. A dotted green vertical line x = −1 is the vertical asymptote between the branches. — Far from the origin, both branches of f(x) = (x² + 3x + 5)/(x + 1) hug the slant asymptote y = x + 2.

How to Find a Slant Asymptote

Compare degrees. A slant asymptote exists only if the degree of the numerator is exactly 1 more than the degree of the denominator (and the fraction is in lowest terms).
Divide the numerator by the denominator. Use synthetic division if the denominator is $x - c$ , and long division otherwise.
Write the result as quotient plus remainder over the denominator.
The slant asymptote is $y =$ the quotient. Ignore the remainder.

Degree of top vs. bottom	End behavior
Top less than bottom	Horizontal asymptote $y = 0$
Equal degrees	Horizontal asymptote $y =$ ratio of leading coefficients
Top is 1 more	Slant asymptote $y =$ quotient
Top is 2 or more more	No line asymptote; the graph follows a curve like a parabola

Worked Examples

Example 1: Synthetic division

Find the slant asymptote of $f(x) = \frac{x^2 + 3x + 5}{x + 1}$ .

Compare degrees: the top is degree 2 and the bottom is degree 1, so there is a slant asymptote

The bottom is $x - (-1)$ , so use synthetic division with $-1$ and coefficients $1, 3, 5$

Bring down 1. Multiply $1 \cdot (-1) = -1$ ; add $3 + (-1) = 2$

Multiply $2 \cdot (-1) = -2$ ; add $5 + (-2) = 3$ , the remainder

f(x) = x + 2 + \frac{3}{x + 1}

Write the result.

Answer: $y = x + 2$ . The graph above also has the vertical asymptote $x = -1$ .

Example 2: A steeper slant

Find the slant asymptote of $f(x) = \frac{2x^2 - 5x - 1}{x - 3}$ .

Degrees are 2 and 1, so there is a slant asymptote

Synthetic division with 3 and coefficients $2, -5, -1$ : bring down 2

Multiply $2 \cdot 3 = 6$ ; add $-5 + 6 = 1$

Multiply $1 \cdot 3 = 3$ ; add $-1 + 3 = 2$ , the remainder

f(x) = 2x + 1 + \frac{2}{x - 3}

Write the result.

Answer: $y = 2x + 1$

Example 3: Long division

Find the slant asymptote of $f(x) = \frac{x^3 - 2x^2 + 4}{x^2 - 1}$ .

Degrees are 3 and 2, so there is a slant asymptote. The bottom is not $x - c$ , so use long division

Divide the leading terms: $\frac{x^3}{x^2} = x$ . Multiply: $x(x^2 - 1) = x^3 - x$

(x^3 - 2x^2 + 0x + 4) - (x^3 - x) = -2x^2 + x + 4

Subtract.

Divide the leading terms: $\frac{-2x^2}{x^2} = -2$ . Multiply: $-2(x^2 - 1) = -2x^2 + 2$

(-2x^2 + x + 4) - (-2x^2 + 2) = x + 2

Subtract (the remainder).

f(x) = x - 2 + \frac{x + 2}{x^2 - 1}

Write the result.

Answer: $y = x - 2$

Example 4: Where the graph crosses its asymptote

Does the graph of $f(x) = \frac{x^3 - 2x^2 + 4}{x^2 - 1}$ from Example 3 ever cross its slant asymptote?

\frac{x + 2}{x^2 - 1} = 0

The graph meets $y = x - 2$ exactly when the remainder part is 0.

x + 2 = 0

A fraction is 0 when its top is 0.

x = -2

Subtract 2 from both sides.

Check that the bottom is not 0 there: $(-2)^2 - 1 = 3$

y = -2 - 2 = -4

Find the point.

Answer: Yes, once, at $(-2, -4)$ . A graph can cross a slant asymptote; it only has to get close to it far out to the left and right.

Example 5: Word problem with average cost

A print shop’s cost to make $x$ posters is $C(x) = 0.5x^2 + 3x + 200$ dollars, so the average cost per poster is $A(x) = \frac{0.5x^2 + 3x + 200}{x}$ . Find the slant asymptote of $A$ , and compare it with $A(100)$ .

A(x) = 0.5x + 3 + \frac{200}{x}

Divide each term by $x$ .

y = 0.5x + 3

The polynomial part is the asymptote.

0.5(100) + 3 + \frac{200}{100} = 50 + 3 + 2 = 55

Find $A(100)$ .

The asymptote at $x = 100$ gives $0.5(100) + 3 = 53$

Answer: $y = 0.5x + 3$ . At 100 posters, the average cost is $55, just $2 above the asymptote, because the fixed $200 is spread over many posters.

Common Mistakes

Mistake	Example of the error	Fix
Using the remainder in the asymptote	$y = x + 2 + 3$	Drop the remainder: $y = x + 2$ .
Looking for a slant asymptote when degrees are equal	Dividing $\frac{x^2 + 1}{x^2 - 4}$ to get a slant line	Equal degrees give a horizontal asymptote, here $y = 1$ .
Wrong number in synthetic division	Using $+1$ for the divisor $x + 1$	Use the zero of the divisor: $x + 1 = 0$ gives $-1$ .
Skipping a missing term	Dividing $x^3 - 2x^2 + 4$ with no $0x$ place	Write $x^3 - 2x^2 + 0x + 4$ so the columns line up.
Thinking the graph can’t cross it	Saying a slant asymptote is never touched	Set the remainder fraction equal to 0 to find any crossing.

Related Methods

Holes first

Before dividing, factor and cancel any common factors. A canceled factor makes a hole, not an asymptote, as shown in graphing rational functions.

Other asymptotes

A rational function with a slant asymptote has no horizontal asymptote, but it can still have vertical asymptotes where the denominator is 0.

Quick check with large x

Plug in a large number like $x = 1000$ . The function value should be close to the asymptote’s value. For Example 1, $f(1000) \approx 1002.003$ and $1000 + 2 = 1002$ .

Practice Problems

Find the slant asymptote of $f(x) = \frac{x^2 + 5x + 2}{x + 2}$ .
Find the slant asymptote of $f(x) = \frac{3x^2 - x + 4}{x - 1}$ .
Find the slant asymptote of $f(x) = \frac{x^2 - 4}{x + 1}$ .
Find the slant asymptote of $f(x) = \frac{2x^3 + x^2 - 3}{x^2 + 1}$ .
Find the slant asymptote of $f(x) = \frac{x^2 + 1}{2x - 4}$ .
Does $f(x) = \frac{x^2 + 1}{x^3 - 8}$ have a slant asymptote?
Does $f(x) = \frac{x^3 + 2}{x - 1}$ have a slant asymptote?
Find all asymptotes of $f(x) = \frac{x^2 - x - 6}{x - 1}$ .
Where does $f(x) = \frac{x^3 + x^2 + 2}{x^2 + 1}$ cross its slant asymptote?
A delivery company’s average cost per mile is $A(x) = \frac{0.02x^2 + 4x + 300}{x}$ dollars for an $x$ -mile route. Find the slant asymptote and compare it with $A(500)$ .

Show answers

Synthetic division with −2 and $1, 5, 2$ : bring down 1; $1(-2) = -2$ , $5 - 2 = 3$ ; $3(-2) = -6$ , $2 - 6 = -4$

x + 3 - \frac{4}{x + 2}