The concept of infinity is one that many struggle to grasp. Many think of it as the largest number possible in math. However, it is important to know that infinity is not a number, but rather a concept that things can grow larger and larger. This concept is essential to understand in order to succeed in certain math classes like calculus. Every part of calculus is directly or indirectly related to infinity. Limits create the foundation for differentiation and integration. The idea of taking the limit of a function draws from the concept of infinity. When taking a limit, you come ever closer to that specified value but you never reach it. In other words, you are going infinitesimally closer to a value without reaching it. Derivatives branch off this idea. They take two points and bring them infinitesimally close to each other until they approach a value which would be the slope. Integration utilizes infinity more directly. The concept of integration is adding up an infinite amount of infinitesimally wide rectangles. Though the limit of the number of rectangles approaches infinity, we get a finite answer for proper integrals. This is one of the biggest concepts many struggle to understand. How is it possible to add up an infinite number of things and get a finite answer? The answer lies in the limit computations that lead to the concept of antiderivatives.
I find the most confusing concept regarding infinity is the idea that some infinities are greater than others. This problem is more prevalent when taking limits of indeterminate forms. For example, two indeterminate forms include ∞ – ∞ and ∞/∞. Logically, you may think that the answers would be 0 and 1 respectively, however, this is not the case. The idea is that both infinities are growing at certain rates. Hence why we incorporate L’Hopital’s rule and take the derivative of the top function over the derivative of the bottom function to reach an answer. As a visual, take the graphs of f(x) = x and g(x) = e^x as x approaches infinity. Both function values of each graph will also approach infinity, however, the value of g(x) will always be greater than that of f(x), therefore, g(x)’s infinity is greater than f(x)’s infinity.
