Oblique or Slant Asymptote

When a Linear Asymptote is not parallel to either x-axis or y-axis, its called Oblique Asymptote. Oblique Asymptote is also called as the slant asymptote. In the below sections there are many solved examples provided for a better understanding. Students can also connect to an online tutor and get the required help with asymptotes.

The general form of a Rational Function is

a_n ≠ 0 and b_m ≠ 0. n is the degree of the numerator and m is the degree of the denominator.

The domain of a rational function is the set of all real numbers except the numbers for which h(x) = 0. For example, for the rational function,

The denominator is zero when x - 1 = 0 that is when x = 1. Hence the domain of R(x) is the set of all real numbers except 1.

If g(x) and h(x) are polynomials of degree n and m respectively and have no common factors, then

Case 1 : If a is a real number such that h(a) = 0, then x = a is a Vertical Asymptote of R(x).

Case 2 : If m = n then y = $\frac{an}{bm}$ is a Horizontal Asymptote of R(x)

Case 3 : If m > n , then y = 0 is a Horizontal Asymptote of R(x)

Case 4 : If n > m, then R(x) has no Horizontal Asymptote.

But if n = m+1, then R(x) has Slant Asymptote. Slant Asymptote is also called as the Oblique Asymptote. That is, if the degree of the numerator of a rational function is exactly one more than the degree of the denominator, then the rational function R(x) has Oblique Asymptote.

For example, consider the function,

The degree of the numerator is 2 and the degree of the denominator is 1. Hence the degree of the numerator is exactly one more than the degree of the denominator. Hence, R(x) has a Slant or Oblique Asymptote

Another way to define an oblique asymptote for a rational function R(x) is,

Since the perpendicular distance between R(x) and the line y = mx+b approaches zero as x becomes unbound.

The function R(x) = `(x^2+1)/x`

The degree of the numerator is exactly one more than the degree of the denominator. R(x) has Oblique Asymptote

The graph of R(x) is shown above. As x tends to negative infinity or as x tends to positive infinity, the graph of R(x) approaches the line y = x. It is the Oblique Asymptote of R(x).

Rational Function R(x) has an Oblique or Slant Asymptote if R(x) is improper and the degree of the numerator is exactly one more than the degree of the denominator.

as x→∞ or as x→ -∞ Hence as x→∞ or as x→ -∞ , R(x) → f(x). This is the Oblique Asymptote if the degree of P(x) is exactly one more than the degree of the denominator Q(x).

An oblique Asymptote is a line of the form y = mx + b where m ≠ 0.

To find Oblique Asymptote of a rational function,

Step 1 : Divide the numerator by the denominator using long division or synthetic division.

Step 2 : The result of division is a non-fractional part and a fractional part. The non-fractional part is the Oblique Asymptote. The fractional part approaches zero as x tends to negative infinity or positive infinity

Below are the examples on asymptote

Example 1: Find the Oblique asymptote of

Solution:

Step 1: Divide the numerator by denominator by using long division so that we get

The result is R(x) =

As x→∞ or as x→ -∞

approaches zero

Hence, as x→∞ or as x→ -∞, R(x) → x+1and hence we can conclude that the function R(x) has an oblique asymptote y = x+1

Example 2: Find the slope of Oblique asymptote of the function

Solution:

To solve this problem, we cannot start the division directly, but we will have to first simplify the expressions. The first two fractions have the same denominator x. So we can add the numerators

This is already in the required form and so we do not have to divide!

As x→∞ or as x→ -∞

approaches zero.

Hence, as x→∞ or as x→ -∞, R(x) → 2x and hence we can conclude that the function R(x) has an oblique asymptote y = 2x.

The slope of line y = mx+b is m. Hence the slope of y = 2x is 2.

The Oblique Asymptote of R(x) is the line y = 2x with a slope of 2.