Understanding Oblique Asymptotes: A Step-by-Step Guide with Examples

oblique asymptote

An oblique asymptote, also known as a slant asymptote, is an asymptote that is not horizontal or vertical

An oblique asymptote, also known as a slant asymptote, is an asymptote that is not horizontal or vertical. It occurs when the degree of the numerator of a rational function is one greater than the degree of the denominator.

To find the equation of the oblique asymptote, you can use long division or synthetic division. Here’s a step-by-step process:

1. Divide the numerator by the denominator. Write the quotient as Q(x) and the remainder as R(x).

2. If the divisor is of degree n, and the remainder is nonzero, then the equation of the oblique asymptote is given by the function Q(x) + R(x)/divisor.

Let’s take an example to illustrate the process:

Consider the rational function f(x) = (x^2 + 2x + 3)/(x – 1).

1. Divide x^2 + 2x + 3 by x – 1.

x + 3
___________
x – 1 | x^2 + 2x + 3

x + 3
___________
x – 1 | x^2 + 2x + 3
– (x^2 – x)
___________
3x + 3
– (3x – 3)
___________
6

So, the quotient Q(x) = x + 3 and the remainder R(x) = 6. Since the degree of the numerator is one higher than the degree of the denominator, we have an oblique asymptote.

2. The equation of the oblique asymptote is Q(x) + R(x)/divisor.

Oblique asymptote: f(x) = x + 3 + 6/(x – 1).

If you graph the function f(x) and the oblique asymptote, you will observe that the function approaches the asymptote as x moves towards positive or negative infinity.

More Answers: