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.
- Horizontal asymptote (HA) - It is a horizontal line and hence its equation is of the form y = k.
- Vertical asymptote (VA) - It is a vertical line and hence its equation is of the form x = k.
- 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) = (3x2 + 6x) / (x2 + 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.
- If both the polynomials have the same degree, divide the coefficients of the leading terms. This is your asymptote!
- If the degree of the numerator is less than the denominator, then the asymptote is located at y = 0 (which is the x-axis).
- If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote!
Example: In the function f(x) = (3x2 + 6x) / (x2 + 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) = (3x2 + 6x) / (x2 + 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:
|Horizontal Asymptote||Vertical Asymptote|
|It is of the form y = k.||It is of the form x = k.|
|It is obtained by taking the limit as x→∞ or x→ -∞.||It is obtained by taking the limit as y→∞ or y→ -∞.|
|It may cross the curve sometimes.||It will never cross the curve.|
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?
The slant asymptote of a rational function is obtained by dividing its numerator by denominator using the long division. The quotient of the division (irrespective of the remainder) preceded by "y =" gives the equation of the slant asymptote. Here is an example.
Example: Find the slant asymptote of y = (3x3 - 1) / (x2 + 2x).
Let us divide 3x3 - 1 by x2 + 2x using the long division.
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:
Example 1: Find asymptotes of the function f(x) = (x2 - 3x) / (x - 5).
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.
Example 2: Can a rational function have both horizontal and oblique asymptotes? Justify your answer.
A rational function has
- 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) = 2x2 - 3x + 7.
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.
FAQs on Asymptotes
What is the Meaning of Asymptotes?
Asymptotes are imaginary lines in the graph of a function to which a part of the curve is very close to but an asymptote never touches the graph. There are 3 types of asymptotes a function can have:
- Horizontal Asymptote (HA)
- Vertical Asymptote (VA)
- Slant Asymptote (Oblique Asymptote)
How to Find Horizontal and Vertical Asymptotes of an Exponential Function?
An exponential function is of the form y = ax + b. Here are the rules to find the horizontal and vertical asymptotes of an exponential function.
- Since an exponential function is defined everywhere, it has no vertical asymptotes.
- As x→∞ or x→ -∞, y → b. Therefore, the horizontal asymptote of y = ax + b is y = b.
We can also draw the exponential graph to identify the asymptotes.
What is an Asymptote in Simple Terms?
An asymptote is a horizontal/vertical/slant line to which the curve is very close to but the curve doesn't touch the asymptote.
What are the Rules to Find Asymptotes?
Here are the rules to find asymptotes of a function y = f(x).
- To find the horizontal asymptotes apply the limit x→∞ or x→ -∞.
- To find the vertical asymptotes apply the limit y→∞ or y→ -∞.
- To find the slant asymptote (if any), divide the numerator by the denominator.
How to Find Horizontal and Vertical Asymptotes of a Logarithmic Function?
A logarithmic function is of the form y = log (ax + b).
- Its vertical asymptote is obtained by solving the equation ax + b = 0 (which gives x = -b/a).
- As x→∞ or x→ -∞, y does not tend to any finite value. Hence it has no horizontal asymptote.
How to Find Oblique Asymptote?
A rational function has an oblique asymptote only when its numerator has a degree just one more than that of its denominator. It is obtained by dividing the numerator by its denominator using the long division of polynomials.
How are Asymptotes Helpful in Graphing Rational Function?
Asymptotes are really helpful in graphing functions as they determine whether the curve has to be broken horizontally and vertically. While graphing, the curve should never touch the asymptotes.
Does Every Rational Function Have a Slant Asymptote?
No, every rational function doesn't have a slant asymptote. A rational function has a slant asymptote only when the degree of its numerator is greater than that of the denominator.