Asymptotes
Engineering Context:
As an engineer, it is important to understand asymptotes of functions in order to understand limits on the applicability of mathematical models. When using an asymptotic function to model a real life situation, the function may have a limited range of physical validity. rational functions, , are frequently used as models in physical situations. All rational functions have at least one vertical asymptote, with many having one or more horizontal asymptotes as well.
MAE: Newton's law of universal gravitation is given by the formula:
ECE: The voltage across a capacitor, assuming a constant voltage source applied to an RC circuit, can be modeled by the differential equation:
BENG: In enzyme kinetics, the rate of an enzymatic reaction is modeled by:
CEE: The rational function model is a mathematical model that is widely utilized in photogrammetry and surveying of geographical terrain. It relates ground points to satellite (or other sensor) points using rational polynomial coefficients. Using this model helps to reduce discrepancies between pixel points and actual data points, and can correct for the coordinate shift that can occur in a remote sensing situation. Any sort of modeling with rational functions requires an understanding of asymptotes in order to fit the model to the situation and to avoid "nuisance asymptotes" that get in the way of data collection or analysis.
The Essentials
An asymptote is a type of restriction on the domain or range of a given function. Graphically, an asymptote looks like a value line that a function gets infinitely close to but never actually reaches.
Asymptotes can be horizontal asymptotes, vertical asymptotes, or oblique asymptotes. Horizontal asymptotes appear as horizontal lines on a graph, vertical asymptotes as vertical lines, and oblique asymptotes as diagonal lines.
The function has a horizontal asymptote at and a vertical asymptote at
Figure 1:
approaches the lines and but never reaches them. and for all values of the function.
The function has an oblique asymptote at as well as a vertical asymptote at :
Figure 2:
For this function, for all values.
Mathematically, a function has a horizontal asymptote if
For example, for :
thus has a horizontal asymptote at .
A function has a vertical asymptote if
in which case the vertical asymptote is defined as . Using the same example of ,
thus has a vertical asymptote at .
A function has an oblique asymptote if
A Deeper Dive
When given a function , horizontal asymptotes can be found by simply taking the limit of the function as approaches . For example, to find the horizontal asymptote of , take and , both of which are equal to 4. Thus, the function has a horizontal asymptote at . If the limit as approaches of the function is equal to , the function has no horizontal asymptotes.
Standard polynomial, trigonometric, and logarithmic equations do not have horizontal asymptotes. Take for example . This function does have a restriction on its range, since all values are in the interval However, these lines of and are not technically horizontal asymptotes, because does not infinitely approach either line, but rather oscillates between them. Asymptotes are just one type of domain and range restriction a function can have.
When given a function , vertical asymptotes can be found by looking for cases where the function is undefined. Frequently, this occurs when pluggin a value of into the equation results in a divisor of zero. For example, Take . This function will always generate an output unless the denominator euqals zero, in which case it will be undefined. Setting and solving for tells us that the function is undefined when , and thus has a vertical asymptote at .
The natural log function is an example of a function that has no vertical asymptote but not a divizor of zero. The natural log function is only defined for numbers greater than zero, and thus only generates an output for positive values of . Because of this, it has a vertical asymptote at . Other logarithmic functions also have vertical asymptotes, all of which occur at the value of for which .
The function has multiple verticle asymptotes. Since the function is equal to , vertical asymptote at every for which . It turns out that four of the six standard trigonometric functions, all except and , have vertical asymptotes. This is because these functions have cases where a denominator equals zero. For example, since , has a vertical asymptote at every for which .
All of these cases of vertical asymptotes can be confirmed by using the limit definition shown above:
Given a rational function , any oblique asymptote can be found by finding the quotient . Dividing the numerator by the denominator will result in a quotient and a remainder : . The oblique asymptote is equal to just the quotient. For example, with the rational function , , , and . Thus, the oblique asymptote of this function is .
Practice
Find all the horizontal and vertical asymptotes of the following functions:
Exercise 1.
Exercise 2.
For Exercise 3, find the oblique asymptote of the rational function:
Exercise 3.
Solutions:
Solution 1.
horizontal:
vertical:
Solution 2.
horizontal: ,
vertical: none
Solution 3.
oblique asymptote: