Calculus: Introduction to Limits

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Limits are usually the first topic covered in a Calculus course. They are the foundation of most higher mathematics and especially for derivatives, which will come next, so it's important to have a firm grasp on the topic.

The limit of a function is a value, L, if we can make the values of f(x) arbitrarily close to L (as close as we wish) by taking x to be sufficiently close to a (on either side of a), but not equal to a.

I know, this is not a very firm definition, but it is the best I've found. We write a limit as follows:

And we say "the limit of f(x) as x approaches a equals L." What this means, then, is that as x gets very close to a (but does not necessarily equal a), f(x) gets very close to L (but again, does not necessarily equal to L). It is important to note that while it may happen that this limit is actually equal to the functional value of f(x) at a (that is, L = f(a)), it is not always the case. In fact, when finding a limit, we are not actually looking at what happens at the point x = a. We are only considering what happens as x approaches or gets closer and closer to a.

You've seen something like this before, probably, in a class like College Algebra. You have probably heard of asymptotes. We will cover this in more detail later, but an asymptote is a limit--as x approaches a vertical asymptote, f(x) approaches either positive or negative infinity. You know that the function never actually crosses this vertical asymptote, so we are only looking at x-values sufficiently close to the asymptote value.

When dealing with limits, how close is "sufficiently" or "very" close? Well, it's as close as you want. Mathematically, we want to consider values infinitely close to a.

In the beginning, you will probably use a table of values to estimate a limit. This is done by choosing x-values on either side of the value in question, and choosing them closer and closer to the value (but never actually choosing the value in question), and finding the f(x) values at those points. You can then see what number y or f(x) is approaching as x approaches the value in question. This is a great way to explore limits in the beginning, but you will soon find it cumbersome and not very accurate.

Somtimes, a limit may be "one-sided," that is, the limit coming from one side of a is different than the limit coming from the other. We write one-sided limits as follows: 

 The first is the right-hand limit of f(x). We say "the limit of f(x) as x approaches a from the right." The second is the left-land limit. In the first case, you are approaching the value a from the right--i.e. from the positive x-value side, and in the second, you are approaching from the left. This seems strange, but there are cases in which these limits may not be the same.

In fact, if the right- and left-hand limits at a particular value are the same, then we say that the limit as x approaches a exists, and is equal to L. Therefore, if the right- and left-hand limits at a particular value are NOT the same, then the limit at that value does not exist, and we write "DNE".

You can use the graph of a function to guesstimate a limit value. For example, estimate the following limit:

Begin by graphing the function inside the limit. On your calculator, you will use the variable x in place of the variable t. It is fine to replace variables in that way. Use the zoom feature to zoom in once. Press ZOOM, then press 2: ZoomIn, then press enter to zoom in at the origin. Here is what you should see.

Now, we are looking for what y-value is approached as x approaches 0. Well, we know that L is between 0 and 1 from looking at the graph. Go over to the Table screen (2ND, then GRAPH) and find x = 0. You'll notice that the y-value given says "ERROR." Press "+", and then type in 0.01 to see the table in increments of 0.01. You may need to arrow up or down to find x = 0 again. Still, it says "ERROR", but notice the values around 0.

Notice that f(-0.02) = 0.16667, as does f(-0.01), f(0.01), and f(0.02). Since these x-values are very close to 0, and since the y-values are changing very little, we can guess that the limit in question (as x approaches 0) is probably equal to 0.16667, which is a rounded decimal approximation of 1/6.

While using the calculator is helpful for finding limits (I use it on more complicated limits if I am struggling), it is not the best way and is not always correct. The next thing we will do is look at limit laws and how to use them to calculate exact values of limits.