Asymptotes

Asymptotes are imaginary lines to which the total graph of a function or a part of the graph is very close. The asymptotes are very helpful in graphing a function as they help to think about what lines the curve should not touch.

Let us learn about asymptotes and their types along with the process of finding them with more examples.

What is an Asymptote?

An asymptote is a line being approached by a curve but never touching the curve. i.e., an asymptote is a line to which the graph of a function converges. We usually do not need to draw asymptotes while graphing functions. But graphing them using dotted lines (imaginary lines) makes us take care of the curve not touching the asymptote. Hence, the asymptotes are just imaginary lines. The distance between the asymptote of a function y = f(x) and its graph is approximately 0 when either the value of x or y tends to ∞ or -∞.

Types of Asymptotes

There are 3 types of asymptotes.

1. Horizontal asymptote (HA) - It is a horizontal line and hence its equation is of the form y = k.
2. Vertical asymptote (VA) - It is a vertical line and hence its equation is of the form x = k.
3. Slanting asymptote (Oblique asymptote) - It is a slanting line and hence its equation is of the form y = mx + b.

Here is a figure illustrating all types of asymptotes.

How to Find Asymptotes?

Since an asymptote is a horizontal, vertical, or slanting line, its equation is of the form x = a, y = a, or y = ax + b. Here are the rules to find all types of asymptotes of a function y = f(x).

• A horizontal asymptote is of the form y = k where x→∞ or x→ -∞. i.e., it is the value of the one/both of the limits lim ₓ→∞ f(x) and lim ₓ→ -∞ f(x). To know tricks/shortcuts to find the horizontal asymptote, click here.
• A vertical asymptote is of the form x = k where y→∞ or y→ -∞. To know the process of finding vertical asymptotes easily, click here.
• A slant asymptote is of the form y = mx + b where m ≠ 0. Another name for slant asymptote is an oblique asymptote. It usually exists for rational functions and mx + b is the quotient obtained by dividing the numerator of the rational function by its denominator.

Let us study more about the process of finding each of these asymptotes in detail in upcoming sections.

How to Find Vertical and Horizontal Asymptotes?

We usually study the asymptotes of a rational function. Of course, we can find the vertical and horizontal asymptotes of a rational function using the above rules. But here are some tricks to find the horizontal and vertical asymptotes of a rational function. Also, we will find the vertical and horizontal asymptotes of the function f(x) = (3x 2 + 6x) / (x 2 + x).

Finding Horizontal Asymptotes of a Rational Function

The method to find the horizontal asymptote changes based on the degrees of the polynomials in the numerator and denominator of the function.

1. If both the polynomials have the same degree, divide the coefficients of the leading terms. This is your asymptote!
2. If the degree of the numerator is less than the denominator, then the asymptote is located at y = 0 (which is the x-axis).
3. If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote!

Example: In the function f(x) = (3x 2 + 6x) / (x 2 + x), the degree of the numerator = the degree of the denominator ( = 2). So its horizontal asymptote is

y = (leading coefficient of numerator) / (leading coefficient of denominator) = 3/1 = 3.

Hence, its HA is y = 3.

Finding Vertical Asymptotes of a Rational Function

To find the vertical asymptote of a rational function, we simplify it first to lowest terms, set its denominator equal to zero, and then solve for x values.

Example: Let us simplify the function f(x) = (3x 2 + 6x) / (x 2 + x).

f(x) = 3x (x + 2) / x (x + 1) = 3(x+2) / (x+1).

When we set denominator = 0, x + 1 = 0. From this, x = -1.

So its VA is x = -1.

Note that, since x is canceled while simplification, x = 0 is a hole on the graph. It means, no point on the graphs corresponds to x = 0.

We can see both HA and VA of this function in the graph below. Also, observe the hole at x = 0.

Difference Between Horizontal and Vertical Asymptotes

Here are a few differences between horizontal and vertical asymptotes:

Slant Asymptote (Oblique Asymptote)

As its name suggests, a slant asymptote is parallel to neither the x-axis nor the y-axis and hence its slope is neither 0 nor undefined. It is also known as an oblique asymptote. Its equation is of the form y = mx + b where m is a non-zero real number. A rational function has an oblique asymptote only when its numerator is exactly 1 more than its denominator and hence a function with a slant asymptote can never have a horizontal asymptote.

How to Find Slant Asymptote?

Hence, y = 3x - 6 is the slant/oblique asymptote of the given function.

Important Notes on Asymptotes:

• If a function has a horizontal asymptote, then it cannot have a slant asymptote and vice versa.
• Polynomial functions, sine, and cosine functions have no horizontal or vertical asymptotes.
• Trigonometric functions csc, sec, tan, and cot have vertical asymptotes but no horizontal asymptotes.
• Exponential functions have horizontal asymptotes but no vertical asymptotes.
• The slant asymptote is obtained by using the long division of polynomials.

☛ Related Topics:

• Asymptote Calculator
• Graphing Functions Calculator
• Graphing Calculator

Asymptotes Examples

Example 1: Find asymptotes of the function f(x) = (x 2 - 3x) / (x - 5). Solution: Finding Horizontal Asymptote: The degree of numerator, d(n) = 2 and the degree of the denominator, d(d) = 1 So d(n) > d(d). Thus, the function has no HA. Finding Vertical Asymptote: The function is in its simplest form. Set denominator = 0. x - 5 = 0
x = 5 So VA is x = 5. Finding the Slant Asymptote: Dividing the numerator by denominator, The oblique asymptote is y = x + 2. Answer: No HA, VA is x = 5, and slant asymptote is y = x + 2.

• an oblique asymptote when its numerator's degree is greater than that of the denominator.
• a horizontal asymptote when its numerator's degree is less than or equal to that of the denominator.

Hence, the same rational function cannot have both oblique and horizontal asymptotes.

Answer: No and the answer is justified.

Example 3: Find the asymptotes of the quadratic function f(x) = 2x 2 - 3x + 7. Solution: A quadratic function is a polynomial and hence it doesn't have any type of asymptotes. This is because f(x) does not tend to any finite number as x tends to infinity (so no HA). Also, f(x) is defined for all real numbers (so no VA). Answer: No asymptotes.

Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.