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/2^{x} 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^{∞}, 0^{0}, 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.