How do you find slant asymptotes?

Steps to Solve

1. Identify the degrees of the polynomials

Check the degree of the numerator and the degree of the denominator of your rational function. Slant asymptotes occur if the degree of the numerator is exactly one more than the degree of the denominator.

2. Perform polynomial long division

Divide the numerator by the denominator using polynomial long division or synthetic division if applicable. The result will be a linear expression plus a remainder.

3. Extract the linear term

The quotient from the polynomial division gives the equation of the slant asymptote. Ignore the remainder term as it becomes insignificant as x approaches infinity.

Example

For the rational function, let’s say ( \frac{x^2 + 3x + 5}{x + 1} ):

• The degree of the numerator ((x^2)) is 2.
• The degree of the denominator ((x)) is 1.
• Since 2 = 1 + 1, we can proceed with the division.

1. Setup the division

Divide (x^2 + 3x + 5) by (x + 1):

$$ x^2 \div x = x $$

2. Multiply & subtract

Multiply (x) by (x + 1):

$$ x \cdot (x + 1) = x^2 + x $$

Subtract (x^2 + x) from (x^2 + 3x + 5):

$$ (x^2 + 3x + 5) - (x^2 + x) = 2x + 5 $$

3. Divide again

Divide (2x) by (x):

$$ 2x \div x = 2 $$

Multiply & subtract again:

$$ 2 \cdot (x + 1) = 2x + 2 $$

Subtract (2x + 2) from (2x + 5):

$$ (2x + 5) - (2x + 2) = 3 $$

4. Form the slant asymptote equation

The quotient of the division is (x + 2). Therefore, the slant asymptote is:

$$ y = x + 2 $$