What is a Limit?
An Essential Tool in Calculus
A limit, for those who have never been so fortunate (or unfortunate, as the case may be) to have taken any courses in advanced mathematics, is an essential first step in calculus, and important in that it represents one of the simplest examples of how calculus can be made useful in real-world, practical situations.
Defining a Limit
When asked to find the limits of a function (like almost everything else in calculus, functions play a key role), the student/mathematician is actually being asked a fairly simple question: What is happening to a function as it approaches a certain point?
To put it a little more visually, imagine a graph of a line on a standard Cartesian Coordinate system (which is just a “normal” graph with an x-axis and a y-axis). The line is drawn at a forty five degree angle, passing through the origin (0,0), and through the points (1,1), (2,2), (3,3), etc. This should be a fairly simple line to picture. Mathematically, this graph represents the function f(x)=x.
If one was to ask about the limit of this function at a certain point, such as the point at which the value of y approaches 2, the question would be put like this: What is the limit of x as f(x) approaches 2?
In other words, what is happening to the value of y as the value of x approaches a value of 2? To find this out, one could just plug in some numbers for values of x at 1.8, 1.9, 1.99, 1.999, 1.9999, etc. Doing this (which is certainly unnecessary in such a simple function as this, one would quickly find that the values of y would be always equal to x. Thus, as x approaches 2, y also approaches 2. That, then, is the answer.
This, of course, is perhaps the simplest of all limit examples, and represents perhaps the easiest problem that a practitioner of calculus would ever be faced with. Unfortunately, functions are rarely this simple, and, consequently, neither are limits.
Common Limit Misconceptions
One of the most common mistakes often made by students who are first learning how to define and solve limits, is to assume that being asked about the limit of a certain function is in some way akin to simply finding the value of a function at that particular point. Thus, in the example above, f(x)=x, when asked to find the limit as x approaches 2, they would simply plug 2 into the equation and find the answer. In this case, of course, the answer would turn out to be right, but please understand that this convenience is truly the exception, rather than the rule.
Consider the case of this same function, f(x)=x, but with the added limitation where x cannot equal 2. In this case, what one would have is simply an arbitrarily small “hole” in the function where both x and y should equal 2. Now, when the limit is attempted, it will no longer be possible to simply enter the value into the equation, because that value doesn’t exist.
Limits, once again, are not (it is difficult to stress this point too much) a question of what a function is doing at any single point. It is a question of what is happening around that given point. What actually exists (or does not exist) at that point is absolutely irrelevant, and should not be considered in the least.
Left and Right Hand Limits
One final thing to point out in this all-too-brief introduction to the subject of limits, is the fact that they can come in two varieties, left-hand and right-hand.
The reason for these varieties should be rather self-explanatory: The left-hand limit is asking a question regarding the behavior of a function as a certain point is approached from the left moving right, and the right-hand limit is asking questions regarding the behavior of a function as it approaches from the right.
Why are these different limits necessary? Once again, because there are some rather screwy functions out there. Some systems of functions even have completely different values at different points of the number line, for instance, a function might say that f(x)=x when x<0, but f(x)=x+2 when x>0. In this case, if one was faced with the question of finding the limit of f(x) as x approached zero, there would be entirely different answers based on which direction the problem was being approached (in this case, the left-hand limit would be 0, while the right-hand limit would be 2).
These are just some first steps to the world of limits, and thus to the world of calculus itself. For more information, please read the Suite101 article on functions, as well.
A Limit Example
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