Calculus for Dopes – It Ain’t That Hard – The Limit Part I

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When Sir Isaac Newton was among other things working out the orbits of the planets in the 1600’s, he realized that the mathematics of the day were simply-well-too simple. Thus he invented a new branch of mathematics called the calculus. What was different about this new math was that it enabled one to deal with “infinitesimals,” or extremely small quantities. In fact, by virtue of these infinitesimals, mathematicians can calculate such things as the exact velocity of a moving body at any particular instant in time, or the exact area of a bizarrely irregular shape. The limit is one of the key aspects of calculus that allows us to do these extraordinary things.

With the Age of Enlightenment ushered in on the back of such giants as Isaac Newton, Adam Smith, and Charles-Louis de Montesquieu, the calculus could not have come at a more apropos time to both cradle and nurture the scientific revolution. This new math once exploited, the world would never be the same: the motion of planetary bodies and the universal law of gravitation could now be adequately explained. Newtonian mechanics would take a stronghold on physics for the next three hundred years until Einstein’s new world of relativity would help explain corrections to motion based on the speed of light. And all this capability and discovery because of something as fundamental as the concept of limit. So what is this fascinating concept all about?

If you think of the dictionary definition of limit as something reached, or something which serves as an absolute border or utmost boundary, then you get a good idea of what the mathematical definition is as well. To think during Newton’s day that this simple concept could harbinger a new age of scientific thought would have been presumptuous at best. However, a new age of thought is exactly what we got. You see the limit allows us to calculate something called the derivative, and this notion is what essentially propels all of the calculus, which has as sub-branches the differential calculus and the integral calculus. From these branches, mathematicians probe all of nature using such tools as ordinary and partial differential equations, indefinite and definite integrals, and, where these become inadequate, other tools like power series and Lagrangian multipliers.

To understand the limit intuitively, let’s look at the following mathematical expression, which is called a function: y = 1/x. What this means is that the dependent variable y is equal to one divided by the independent variable x. The values of y are determined by the values we allow x to assume. Thus if x = 10, then y = 1/10. If x = 100 then y = 1/100 and so on. The limit of a function is the value that is reached when the independent variable-in this case x-gets closer and closer, without necessarily equaling, some other value.

For example, in the case of y = 1/x, if we examine what happens to y when x gets closer and closer to 2, we will see that the value of the function, or y, will get closer and closer to ½. We can see this by letting x = 2.01, then 2.001, then 2.0001, and calculating the value for 1/x or y. For each of those values, we will get y = 0.4975, then y = 0.49975, the y = 0.499975. Notice that as x gets closer and closer to 2, then y gets closer and closer to ½ or 0.5.

Now you may say, “What’s the big deal? That’s so obvious!” Yet there are times, when the result is not obvious at all. This is when things get interesting and when we need some very clever tools to figure out what the limit actually is. The limit also helps us talk about things that are mathematically impossible to do. For instance, you learned that you can never divide by zero, and that anything divided by zero was undefined. Yet the concept of limit allows us to talk about this very thing. How so?

Take the function that we have used in this article, y = 1/x. We cannot let x = 0 because we can never calculate the value of y. But now with the concept of limit, we can ask, “What is the limit of the function y as x approaches 0?” If you start to let x equal values that get closer and closer to 0, you will start to see exactly what happens to y. You can then take a good guess as to what happens to y when x gets close to zero. At this point, we have entered a new realm, or should we say, dimension: the one of infinity. And within this realm, a lot of strangely marvelous things occur.

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AUTOPOST by BEDEWY VISIT GAHZLY