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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, $f (x) = \frac{p (x)}{q (x)}$ , 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:

F = G \frac{m_{1} m_{2}}{r^{2}}

where

F

is the force between two objects,

m_{1}

and

m_{2}

are the respective masses of the two objects,

r

is the disctance between their centers of mass, and

G

is the gravitational constant. This equation has a vertical asymptote at

r = 0

. As

r

goes to zero, the modeled gravitational force goes to infinity.

ECE: The voltage across a capacitor, assuming a constant voltage source applied to an RC circuit, can be modeled by the differential equation:

R C \frac{d v (t)}{d t} + v (t) = v_{s}

where

R

is the resistance,

C

is the capacitance,

v

is the voltage across the capacitor, and

v_{s}

is the voltage applied to the circuit. This equation has a horizontal asymptote at

v (t) = v_{s}

meaning that the maximum voltage across the capacitor is approximately equal to the voltage applied to the circuit. If we graph the solution

v

to this differential equation, it will have a horizontal asymptote at

v = 0

BENG: In enzyme kinetics, the rate of an enzymatic reaction is modeled by:

\frac{d [P]}{d t} = \frac{k_{2} {[E]}_{0} [S]}{K_{m} + [S]}

where

\frac{d [P]}{d t}

is the change in product concentration over time,

[E]

is the concentration of the enzyme,

[S]

is the concentration of the substrate (reactant), and

k_{2}

and

K_{m}

are constants. As

[S]

goes to infinity, the reaction rate approaches the horizontal asymptote

V_{\max}

, which is considered the maximum rate of the enzymatic reaction.

V_{\max} = lim_{[S] \to \infty} \frac{k_{2} {[E]}_{0} [S]}{K_{m} + [S]}

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 $f (x) = \frac{1}{x}$ has a horizontal asymptote at $y = 0$ and a vertical asymptote at $x = 0$

$Vertical Asymptote$

Figure 1: $f (x) = \frac{1}{x}$

$f (x) = \frac{1}{x}$ approaches the lines $x = 0$ and $y = 0$ but never reaches them. $x \neq 0$ and $f (x) \neq 0$ for all values of the function.

The function $f (x) = x + \frac{1}{x}$ has an oblique asymptote at $y = x$ as well as a vertical asymptote at $x = 0$ :

$Oblique Asymptote$

Figure 2: $f (x) = x + \frac{1}{x}$

For this function, $x \neq f (x)$ for all values.

Mathematically, a function $f (x)$ has a horizontal asymptote if

lim_{x \to + \infty} f (x) = C

lim_{x \to - \infty} f (x) = C

and

f (x) \neq C

for all

x

values, with

C

being a constant

\neq \pm \infty

. The horizontal asymptote of such a function is the line

f (x) = C

For example, for $f (x) = \frac{1}{x}$ :

lim_{x \to + \infty} \frac{1}{x} = 0

and

lim_{x \to - \infty} \frac{1}{x} = 0

thus $f (x) = \frac{1}{x}$ has a horizontal asymptote at $f (x) = 0$ .

A function $f (x)$ has a vertical asymptote if

lim_{x \to a^{+}} f (x) = \pm \infty

lim_{x \to a^{-}} \frac{1}{x} = \pm \infty

in which case the vertical asymptote is defined as $x = a$ . Using the same example of $f (x) = \frac{1}{x}$ ,

lim_{x \to 0^{+}} \frac{1}{x} = \infty

and

lim_{x \to 0^{-}} \frac{1}{x} = - \infty

thus $f (x) = \frac{1}{x}$ has a vertical asymptote at $x = 0$ .

A function $f (x)$ has an oblique asymptote if

lim_{x \to \infty^{+}} f (x)

lim_{x \to \infty^{-}} f (x)

approaches

m x + b

Oblique asymptotes usually appear with rational functions, and a function will only have an oblique asymptote if the degree of the numerator is exactly 1 more than the degree of the denominator.

A Deeper Dive

When given a function $f (x)$ , horizontal asymptotes can be found by simply taking the limit of the function as $x$ approaches $\pm \infty$ . For example, to find the horizontal asymptote of $f (x) \frac{4 x}{x - 5}$ , take $lim_{x \to \infty^{+}} \frac{4 x}{x - 5}$ and $lim_{x \to \infty^{-}} \frac{4 x}{x - 5}$ , both of which are equal to 4. Thus, the function has a horizontal asymptote at $f (x) = 4$ . If the limit as $x$ approaches $\pm \infty$ of the function is equal to $\pm \infty$ , the function has no horizontal asymptotes.

Standard polynomial, trigonometric, and logarithmic equations do not have horizontal asymptotes. Take for example $f (x) = sin (x)$ . This function does have a restriction on its range, since all values are in the interval $[-1, 1]$ However, these lines of $y = 1$ and $y = -1$ are not technically horizontal asymptotes, because $sin (x)$ 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 $f (x)$ , vertical asymptotes can be found by looking for cases where the function is undefined. Frequently, this occurs when pluggin a value of $x$ into the equation results in a divisor of zero. For example, Take $f (x) = \frac{x + 5}{x + 3}$ . This function will always generate an output unless the denominator euqals zero, in which case it will be undefined. Setting $x + 3 = 0$ and solving for $x$ tells us that the function is undefined when $x = -3$ , and thus has a vertical asymptote at $x = -3$ .

The natural log function $f (x) = ln (x)$ 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 $x$ . Because of this, it has a vertical asymptote at $x = 0$ . Other logarithmic functions also have vertical asymptotes, all of which occur at the value of $x$ for which $f (x) = {log}_{b} (0)$ .

The function $f (x) = tan (x)$ has multiple verticle asymptotes. Since the $tan (x)$ function is equal to $\frac{sin (x)}{cos (x)}$ , $tan (x)$ vertical asymptote at every $x$ for which $cos (x) = 0$ . It turns out that four of the six standard trigonometric functions, all except $sin (x)$ and $cos (x)$ , have vertical asymptotes. This is because these functions have cases where a denominator equals zero. For example, since $csc (x) = \frac{1}{sin (x)}$ , $f (x) = csc (x)$ has a vertical asymptote at every $x$ for which $sin (x) = 0$ .

All of these cases of vertical asymptotes can be confirmed by using the limit definition shown above:

lim_{x \to 0^{+}} \frac{x + 5}{x + 3} = \infty

and

lim_{x \to {-3}^{-}} = - \infty

lim_{x \to 0^{+}} ln (x) = - \infty

lim_{x \to arccos {(0)}^{+}} tan (x) = \infty

and

lim_{x \to arccos {(0)}^{-}} tan (x) = - \infty

Given a rational function $f (x) = \frac{p (x)}{q (x)}$ , any oblique asymptote can be found by finding the quotient $\frac{p (x)}{q (x)}$ . Dividing the numerator $p (x)$ by the denominator $q (x)$ will result in a quotient $Q$ and a remainder $R$ : $\frac{p (x)}{q (x)} = Q + \frac{R}{q (x)}$ . The oblique asymptote $a (x)$ is equal to just the quotient. For example, with the rational function $f = \frac{x^{2} + 1}{x}$ , $p (x) = x^{2} + 1$ , $q (x) = x$ , and $\frac{p (x)}{q (x)} = x + \frac{1}{x}$ . Thus, the oblique asymptote of this function is $a (x) = x$ .

Practice

Find all the horizontal and vertical asymptotes of the following functions:

Exercise 1. $f (x) = \frac{x^{2} + 3}{x^{3} - 1}$

Exercise 2. $f (x) = ln (x^{2} - 4)$

For Exercise 3, find the oblique asymptote of the rational function:

Exercise 3. $f (x) \frac{x^{2} + 2}{x - 1}$

Solutions:

Solution 1.

horizontal: $x = 1$

vertical: $f (x) = 0$

Solution 2.

horizontal: $x = 2$ , $x = -2$

vertical: none

Solution 3.

oblique asymptote: $f (x) = x + 1$

Asymptotes

Engineering Context:

The Essentials

A Deeper Dive

Practice

More Resources