What are Limits in Calculus?

Back when I was taking my introductory calculus courses, one of the first topics that we covered was a section on limits and rates of change.  To a math student, "rate of change" kind of makes some sense.  It sounds like a math concept that explains how fast something is changing.  It's a fairly intuitive concept, and we will eventually see that it is a very important concept as well!  On the other hand, at first glance, the concept of a "limit" sounds a bit more abstract.  We all know about limits in terms of, for example, the speed limit for a car or other defined measures.  But how does this construct fit into calculus, or does "limit" mean something else entirely in this context?

The concept of a limit is essentially the same as when we're talking about a speed limit.  Let's say there is a posted speed limit of 50 mph.  Your car is not allowed to travel any faster than this.  So, imagine then what a graph would look like of the speed of your car over time.  At time = 0, you start stationary.  Then you accelerate and start driving a bit faster, until you get to the speed limit, at which point you don't go any faster (because we obey the laws!), and then you approach a stop sign and slow down to come to a stop.  When you look at the graph of what has just happened, you can see that the speed limit is basically a speed value that your car approaches.

In calculus, the concept of a limit is very similar, yet there is a slight difference that is very important.  In quite basic terms, a limit of a graph (or more accurately, a function) is a value that the graph approaches.  This concept is quite different from asking "what is the value at a point on the graph" because you may have a situation where the graph is non-continuous.  Here are a couple of examples that may clear up this concept of calculus limits.

For the first example, consider the following curve.  It is a smooth, continuous curve, meaning that there are no gaps or sudden jumps in the line.  (I will do a separate post on continuity of graphs in more detail soon.)  The limit of the graph as you move along the line towards the point P (getting infinitely closer but never actually reaching), from either direction, approaches the line of y = 4.  In this case, the graph actually does have a point at y = 4, because it is continuous.  So we can technically say that the limit as f(x) approaches point P is 4, and also that f(x) = 4.

Now, let's consider a discontinuous graph to see the importance of limits.  Recall what a function actually is, and specifically in this case think about piecewise functions, and also perhaps a refresher of the vertical line test would do well here as well.  In the below discontinuous graph, we can see that it is indeed a function because it passes the vertical line test.  It also appears to be a piecewise function, in that there appears to be a different expression that defines different parts of the graph (e.g. the curve, and also the point at P).  Note that on the curve, the point P is shown with an open circle to indicate all values that approach P, but not P.  P is elsewhere on the graph.  Similarly to the continuous expression above, as the curve of the graph approaches point P and gets infinitely closer without ever reaching it, it appears to be heading towards (i.e. the limit of the graph is) 4.  However, the expression f(x) itself clearly is not the same value exactly at P, and in this case we can see that when x = p, y actually equals 10 (while the limit is 4).

I know this has been a very basic explanation of limits, but I only intend for it to serve as an introduction to this calculus concept.  I will be posting additional material on other limit-related topics, including the correct limit notation, the concepts of one-sided limits, continuity of graphs, and asymptotes, and I will present some more introductory posts that discuss tangents, velocity, and rates of change.  And of course, what is a mathematical concept without having a set of laws to go with it?  So I will present to you, for your viewing pleasure, the Limit Laws.  I hope this introduction to limits has been enough to provide you with a basic understanding of what a limit of a function is, and how it differs from a regular expression that you are used to by now.  You can't think of these the same way as you have been regarding expressions up to now.  Before, you wanted to know what was happening AT a point.  With limits, you want to know what is happening AS YOU APPROACH a point.  They sound very similar, but they are distinctly different.  A good understanding of limits is essential for you to be able to work with other calculus problems and concepts.  I hope my coming posts will be helpful for you.  As always, please remember to +1 this if it helped you.

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