## When 0/0 or ∞/∞ Can Have a Value That Makes Sense

In a previous post, Why Division by Zero is Forbidden, I explained why a nonzero number divided by zero is undefined and why zero divided by zero gives us infinitely many answers.

Several comments from math teachers indicate that it is not easy to get this concept across to younger students so that they have a real understanding of the concept. I fully agree. From personal experience, I have found that students initially attain a rudimentary understanding of a math concept, but only time, practice working with the concept, and increased mental maturity can students gain a deeper understanding of a concept.

One reader, Ali-Carmen Houssney, wondered why an expression of the form 0 / 0 is called “indeterminate.” The purpose of this post is to discuss the indeterminate forms of the type 0 / 0 and ∞ / ∞. There are other indeterminate forms which you can look up online, but the forms 0 / 0 and ∞ / ∞ are the two major indeterminate forms that appear in calculus text books.

To get started, I need to discuss the concept of a limiting value of a function f(x) at some specific point x = k. The limit concept is a core concept in differential calculus. The limiting value of a function when x = k is simply a number or value that f(x) gets arbitrarily close to when x gets arbitrarily close to k or +∞.  The definition of a limit of a function only requires that the input variable x gets arbitrarily close to some constant k and not necessarily equal to k. In most cases the limiting value of a function as x gets arbitrarily close to k is just f(k) itself. In other cases when f(k) results in the indeterminate form 0 / 0 or ∞ / ∞, the limiting value of f(x) when x gets arbitrarily close to k may exist or it may not exist. If the limiting value does exist, the limiting value will be a single finite real number or + ∞ which means that the function is increasing/decreasing without any upper/lower boundary. The text boxes and companion graphs below illustrate the limit concept for the functions y = f(x) = Sin(x) / x, y = g(x) = x/2x and y = h(x) = Floor(x). All function outputs in the examples are rounded to 9 decimal places.

What’s important to notice in the examples below, is that values of x very close to k were selected to test whether or not the corresponding function values are very close to some fixed constant L, the limit of the function at x = k. A mathematically rigorous demonstration of the existence of the limit at x = k would show that there is always an open interval (a, b) about k in which all x in (a, b), except x = k, so that the corresponding function values are arbitrarily close to L, the limiting value of the function at x = k.

From a geometric point of view, it’s interesting to see why the limiting value of Sin(θ) / θ = 1 as θ approaches 0. Refer to graph D below. If central angle θ is in radians, the length of the arc on the unit circle that subtends angle θ equals θ units. Now imagine what happens as θ approaches 0; central angle θ approaches 0, Sin(θ) and length of the arc get closer and closer to each other. Hence, Sin(θ) / θ approaches 1 as θ approaches 0. This fundamental limit is used to derive the formulas for the derivative of all trigonometric functions. There is another interesting fact. If θ is converted to degrees, the limiting value of Sin(θ) /θ equals π/180 ≈ 0.017453293. You can easily convince yourself that this is true by setting the angle mode of your calculator to degrees and entering values of θ in the expression Sin(θ) /θ that are very close to 0.

In closing this post, I will mention L’Hôpital’s rule which is a very useful theorem for finding the limiting value of a function f(x) at x = k when f(k) results in the indeterminate form 0 / 0, ∞ / ∞, (-∞) / ∞, ∞ / (-∞) or (-∞) / (-∞).  When f(k) results in indeterminate forms such as 0 * ∞, 1, 00, or ∞ – ∞, calculus students learn techniques for rewriting the expression for f(x) so that L’Hôpital’s rule can be applied.  In order to use L’Hôpital’s rule, one needs to know how to find the derivative of a function.

## Comparing Circular Trig Functions with Hyperbolic Trig Functions

This post does a geometric compare and contrast of the circular functions Cos(x) and Sin(x) with the hyperbolic trig functions Cosh(x) and Sinh(x). I will do this by showing points of the form (Cos(t), Sin(t)) are points on the unit circle, and by showing points of the form (Cosh(t), Sinh(t)) are points on the unit hyperbola.

I will start the discussion by doing a quick review of radian angle measure and the definitions of the circular trigonometric functions Cos(θ), Sin(θ), and Tan(θ) where the input variable θ is the radian measure of a central angle of a circle with center at (0, 0) and radius equal to r. The text box below gives the definitions of the circular functions Cos(θ), Sin(θ), and Tan(θ) along with the radian measure of an angle and the formula for the area of a sector of a circle for angle θ. The formula for the area of a sector of a circle follows from the definition of radian angle measure and the fact that the area of a circular sector is proportional to the length of the arc that subtends the central angle.

The graph below illustrates the circular trig relationships in the text box above when r = 25 units, θ = α = 1.2870 radians and θ = β = -2.8578 radians. α, β, arc lengths and areas of circular sectors are rounded to 4 decimal places.

When the circular trig functions are defined in terms of the unit circle, r = 1, the input variable of the circular trig functions can be treated as an angle with angle measure = t radians, arc length = t length units, or time = t time units. This makes it possible to model periodic motion and periodic processes with circular trig functions. The text box below gives the definitions of Cos(t), Sin(t), and Tan(t) in terms of the unit circle. The unit circle graph illustrates the unit circle relationships when t = 2.58 units and t = -1.6 units. Notice that the area of the circular sector of arc length t = |t|/2 units2.

It’s now time to take a look at the unit hyperbola and the hyperbolic trig functions x = Cosh(t) and y = Sinh(t). My previous post discussed the derivation, the graphs, and some identities for hyperbolic trig functions. If you have not read that post, I strongly suggest that you read it before continuing. The text box below summarizes the key points that are pertinent to this discussion. The graph illustrates the unit hyperbola relationships when t = 2.2 and t = -1.6. An understanding of integral calculus is required to understand the derivation of some hyperbolic relationships.

If you are interested in Einstein’s theory of special relatively, you will find The Geometry of Special Relatively by Tevian Dray a must read. The author shows an elegant way to express the Lorentz transformations in terms of Cosh(β) and Sinh(β) where β is implicitly defined by the equation Tanh(β) = v/c where v is some constant velocity and c is the speed of light. To my amazement, I learned that the Lorentz transformations are just hyperbolic rotations!

Dray’s book inspired me to think more deeply about the circular trig functions, hyperbolic trig functions, and the difference between the two geometries. Dray warns the reader that hyperbola geometry should not be confused with hyperbolic geometry which is the curved geometry of the two-dimensional unit hyperboloid. I will conclude this post with some of my personal observations about circular trig functions and hyperbolic trig functions, and finish with a wonderful example from Dray’s book that really clarified for me the difference between Euclidean geometry and hyperbola geometry.

The diagram below shows a right triangle in hyperbola geometry. The results are astonishing when you consider the definitions of Cos(θ), Sin(θ) and Tan(θ) for Euclidean right triangles. Of course, the triangle below would be impossible in Euclidean geometry, but possible in hyperbola geometry. Just think of the diagram as a clever way to picture time and space quantities in spacetime physics. In hyperbola geometry, a triangle is a right triangle if and only if the lengths of the sides satisfy the Pythagorean Theorem of hyperbola geometry, a2 – b2 = c2 where a = c*Cosh(β), b = c*Sinh(β), and β is an angle of the triangle. From the formulas for the inverse hyperbolic trig functions, it follows that β = arcCosh(5/4) = arcSinh(3/4) = arcTanh(3/5) = 0.693147181. Notice that the hypotenuse of the hyperbolic right triangle is not the longest side.

## Even and Odd Functions

The concepts of even and odd functions are usually introduced in advanced high school algebra, college algebra, trigonometry, or precalculus courses. Trig students learn how to apply the concepts of even and odd functions to simplify trigonometric expressions. Calculus students learn how to apply the concepts of even and odd functions to simplify the calculation of a definite integral. When students understand and can recognize even and odd functions, they are usually amazed how often these functions appear in application problems.

This post will discuss even and odd functions from both an algebraic and geometric point of view. Readers are encouraged to download my free handout Even and Odd Functions which is a handy reference that teachers can give to students.

I will start the discussion by describing even functions. The text box below gives a description of an even function from several points of view. Graph A illustrates what we mean when we say that a graph has symmetry with respect to the y-axis.

The text box below gives a description of an odd function from several points of view. Graph B illustrates what we mean when we say that a graph has symmetry with respect to the origin.

The text box below gives some basic observations about even and odd functions. My free handout Even and Odd Functions gives a more in depth list of the important properties of these functions.

The text box below shows the infinite Taylor series expansion of the function y = Cos(x). Graph C shows the graph of y = Cos(x) and the graph of the first five terms of the Taylor series expansion of Cos(x). Notice that the Taylor series expansion of Cos(x) is the sum and difference of even functions!

The text box below shows the infinite Taylor series expansion of the function y = Sin(x). Graph D shows the graph of y = Sin(x) and the graph of the first five terms of the Taylor series expansion of Sin(x). Notice that the Taylor series expansion of Sin(x) is the sum and difference of odd functions!

I will end this post by showing you how to create an even or odd function from any function y = f(x) that’s not necessarily even or odd. The text box below shows how and why this can be done. Graph E shows the results of creating an even function and an odd function from the function y = f(x) = 0.25(x – 4)2 – 3Sin(x – 4) – 5. This result explains why the hyperbolic functions cosh(x) and sinh(x) are even and odd functions respectively.

More useful tools from Math Teacher’s Resource

•   The graphs in this post were created with my software, Basic Trig Functions. I think that you will find it very useful for teaching mathematical concepts in your classroom and developing custom instructional content. Check it out at mathteachersresource.com/trigonometry.

•   In addition to the Even and Odd Functions Handout linked in this post, Math Teacher’s Resource offers a wide variety of free math handouts, lessons, and exercises available at mathteachersresource.com/instructional-content. All content is available for immediate download. No sign-up required; no strings attached!

## Modeling Limited Population Growth with the Logistic Function

Because of limits on food, living space, disease, current technology, war, and other factors, most populations have limited growth as opposed to unlimited exponential growth which is modeled by the classic exponential growth equation P = P0bt/k. A limited growth population starts growing almost exponentially, but reaches a critical point in time where its growth rate slows, and the population starts to asymptotically approach an upper limit as time increases. There are several models that are used to describe limited growth of a population.

In this post, I will discuss the logistic function which was used by the Belgian mathematician Pierre Francois Verhulst (1804-1849) to study limited population growth. The logistic function also has applications in artificial neural networks, biology, chemistry, demography, ecology, economics, biomathematics, geoscience, mathematical psychology, sociology, political science, probability, and statistics.

The two text boxes below describes the key parameters and relationships between the parameters of a logistic function. Graph (A) shows a typical logistic function curve and how equation parameters can be calculated from known characteristics of the population. If pmax, p0, and tc are known, then a, b, and k can be calculated. Likewise, if a, b, and k are known, then pmax, p0, and tc can be calculated. Keep in mind that all limited growth models can only give us a good approximation of a population value at some point in time.

In most cases, the key parameters of a logistic equation are unknown, but an observed set of data-pairs is known. The least-squares logistic equation of a data set is the best of all possible logistic equations that describes the relationship between the data-pair variables. Best possible equation means that the sum of the squared errors (difference between observed value and predicted value) is minimized. Modern graphing calculators have the capability of findings a least-squares equation for a variety of models such as linear, quadratic, cubic, quartic, sinusoidal, log, exponential, and logistic. When given a logistic type data set, I will use a graphing calculator to find the least-squares logistic equation of the data set, and then calculate various characteristics and properties of the resulting logistic model. I will now take a look at two problems that illustrate how the logistic function can be used to describe limited population growth.

Problem 1 solution: Use math software to do a scatter plot of the data, find the least-squares logistic equation p = 12.0121 / (1 + 10.6694e-0.023856x) of the data set, and then do the appropriate calculations. Refer to graph (B) below. From graph (B) we see that the world population growth rate started to slow in 1999, and the upper limit of the world population is about 12 billion. Keep in mind that this least-squares equation is our current best description of world population growth. Future unknowable events will alter this model.

Problem 2: The logistic function N(t) = 3,600 / (1 + 29.4e-0.2t ) models the spread of a disease in a town. N(t) = the total number of people infected at time t, and t = the number of days after the first reported infections.

(a) How many people were initially infected?

(b) How many people were infected after 10 days and after 30 days?

(c) When did the rate of infection start to slow?

(d) What is the upper limit of the number of infected people?

Problem 2 solution: Use math software to graph the equation, and then do the appropriate calculations. Refer to graph (C) below.

(a) About 118 people were initially infected.

(b) After 10 and 30 days, 723 and 3,355 were infected.

(c) About day 17, the infection rate started to slow.

(d) The upper limit of the number of people infected = 3,600.

Comments:

• It’s fun and interesting to experiment with different logistic function parameters. Experimentation always gives a better learning experience.
• With my graphing calculator, it took about 8 seconds to compute the parameters of a logistic equation. This is an indication of the complexity of the algorithms for computing the parameters of a least-squares equation. I tell my students that they should be ever thankful that they have access to such wonderful computation tools.
• Computer math software allows students to focus on math concepts, and not get lost in gory computational details. This is why graphing calculators have revolutionized the way we teach statistics. Just getting the ‘answer’ is no longer sufficient. Students must be able to interpret and explain the meaning of the answer in the context of the problem.