How to find Vertical Asymptotes?

There are two steps to be followed in order to ascertain the vertical asymptote of rational functions. These are:

Step I: Reduce the given rational function as much as possible by taking out any common factors and simplifying the numerator and denominator through factorization.

Step II: Equate the denominator to zero and solve for x. The value(s) of x is the vertical asymptotes of the function.

Related Articles:

• Curve Tracing
• Rectangular Hyperbola
• Hyperbola

Sample Problems: How to find Vertical and Horizontal Asymptotes?

Problem 1. Find the horizontal and vertical asymptotes of the function: f(x) = [Tex]\frac{x^2-3x}{x+5}  [/Tex].

Solution:

Horizontal Asymptote:

Degree of the numerator = 2

Degree of the denominator = 1

Since the degree of the numerator is greater than that of the denominator, the given function does not have any horizontal asymptote.

Vertical Asymptote:

Since the function is already in its simplest form, just equate the denominator to zero to ascertain the vertical asymptote(s).

⇒ x + 5 = 0

⇒ x = −5

Problem 2. Can a quadratic function have any asymptotes?

Solution:

A quadratic function is a polynomial, so it cannot have any kinds of asymptotes. Since the polynomial functions are defined for all real values of x, it is not possible for a quadratic function to have any vertical asymptotes. Also, since the function tends to infinity as x does, there exists no horizontal asymptote either. 

Problem 3. Find the horizontal and vertical asymptotes of the function: f(x) = [Tex]\frac{3x^2+6x}{x^2+x}  [/Tex].

Solution:

Horizontal Asymptote:

Degree of the numerator = 2

Degree of the denominator = 2

Since the degree of the numerator is equal to that of the denominator, the horizontal asymptote is ascertained by dividing the leading coefficients.

⇒ HA = 2/2 = 1

Vertical Asymptote:

The function needs to be simplified first. [Tex]\frac{3x^2+6x}{x^2+x}=\frac{3x(x+2)}{x(x+1)}=\frac{3(x+2)}{x+1} [/Tex]

Now that the function is in its simplest form, equate the denominator to zero in order to determine the vertical asymptote.

⇒ x + 1 = 0

⇒ x = −1

Problem 4. Find the horizontal and vertical asymptotes of the function: f(x) = 10x² + 6x + 8.

Solution:

The given function is quadratic. A quadratic function is a polynomial, so it cannot have any kinds of asymptotes. Since the polynomial functions are defined for all real values of x, it is not possible for a quadratic function to have any vertical asymptotes. Also, since the function tends to infinity as x does, there exists no horizontal asymptote either. 

Problem 5. Find the horizontal asymptote of the function: f(x) = 9x/x²+2.

Solution:

Degree of numerator = 1

Degree of denominator = 2

Since the degree of the numerator is smaller than that of the denominator, the horizontal asymptote is given by: y = 0.

Problem 6. Find the horizontal and vertical asymptotes of the function: f(x) = x+1/3x-2.

Solution:

Horizontal Asymptote:

Degree of the numerator = 1

Degree of the denominator = 1

Since the degree of the numerator is equal to that of the denominator, the horizontal asymptote is ascertained by dividing the leading coefficients.

⇒ HA = 1/3

Vertical Asymptote:

The function is in its simplest form, equate the denominator to zero in order to determine the vertical asymptote.

⇒ 3x – 2 = 0

⇒ x = 2/3

Problem 7. Find the horizontal and vertical asymptotes of the function: f(x) = x²+1/3x+2.

Solution:

Horizontal Asymptote:

Degree of the numerator = 2

Degree of the denominator = 1

Since the degree of the numerator is greater than that of the denominator, the given function does not have any horizontal asymptote.

Vertical Asymptote:

Since the function is already in its simplest form, just equate the denominator to zero to ascertain the vertical asymtptote(s).

⇒ 3x + 2 = 0

⇒ x = −2/3

To identify the vertical asymptotes of a function, set the denominator equal to zero and solve for x. Since the denominator is factored, set each factor equal to zero and solve individually. To determine the horizontal asymptotes, compare the degrees of the numerator and the denominator.

Asymptotes are important in the study of functions as they provide insights into the long-term behavior of the function and help in understanding its limits as the independent variable approaches certain values or infinity. They are often used in calculus, algebra, and other areas of mathematics to analyze functions and their properties. In this article, we will learn how to find Horizontal and Vertical Asymptotes of any curve.