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Section 1.5 Introduction to limits

We have now motivated the idea of a limit using the two central problems in calculus. Having set the stage, we will dig into defining what we mean by a limit and explore some strategies for computing them.

Our notion of a limit right now is a little bit vague - we think about where a function tends to as we approach a particular number. While we eventually will make this precise, our first step is to build out our intuition.

Let's look at the three examples in Figure 1.5.1 from Figure 2.12 in our text. These three functions all have something funny going on at \(x=2\text{.}\) The function

\begin{equation*} f(x)=\frac{x^2-4}{x-2} \end{equation*}

is not defined at \(x=2\text{,}\) if we try to plug in that value we get zero in both the numerator and denomenator which does not yield a number. The second function

\begin{equation*} g(x)=\frac{|x-2|}{x-2} \end{equation*}

is positive for values of x larger than 2, negative for those less than 2, and also yields zero in the numerator and denomenator at exactly 2. And the last,

\begin{equation*} h(x)=(x-2)^{-2} \end{equation*}

is not defined at two either but only has a zero in the denominator if we try to evaluate at 2.

Figure 1.5.1. The graphs of three functions that are not defined at \(x=2\)

The graphs show us a little bit more about what is going on. The first function simply has a hole at \(x=2\text{,}\) we can imagine just filling it in so that the function is defined there. The second has a jump at \(x=2\text{:}\) the values to the left of two lie on the line \(y=-1\) while those to the right lie on the line \(y=1\text{.}\) The third graph grows quickly near two and leaves the top of the graph.

These are all very different behaviors - and there are even more possiblities with other functions - and taking the limit is key to understanding the differences. Looking at the graphs, what might we conclude about the limit of these functions as \(x\) tends to two. For the first, it looks like the hole occurs at \(y=4\text{.}\) The second doesn't have a limit, but maybe two - one from the left, \(-1\text{,}\) and one from the right, \(1\text{.}\) The last one is perhaps the most puzzling as we can't quite see what is going on. If we zoomed out this picture to include more of the vertical axis, the picture doesn't change much except that the arcs on either side of \(x=2\) move closer and closer together but never touch. So, one way we might describe this is that the function tends to positive infinity at this point.

In our discussion of the tangent and area problems, we saw that seemingly very specific problems generalized to very large classes of problems. We'd like to do that here as well - we want a notion of a limit that allows us to solve lots of types of limit problems without having to analyze each function individually as they arise. To facilitate this, we introduce both notation and two related general techniques.

First, let's try to make a definition of the limit that's a little more precise and that will give us some short-hand for talking about them.

Definition 1.5.2.
Let \(f(x)\) be a function defined at all values in an open interval containing \(a\text{,}\) with the possible exception of a itself, and let \(L\) be a real number. If all values of the function \(f(x)\) approach the real number \(L\) as the values of \(x\) (that are not equal to \(a\)) approach the number \(a\text{,}\) then we say that the limit of \(f(x)\) as \(x\) approaches a is \(L\text{.}\) (More succinct, as \(x\) gets closer to \(a\text{,}\) \(f(x)\) gets closer and stays close to \(L\text{.}\)) Symbolically, we express this idea as
\begin{equation*} \lim_{x \rightarrow a}f(x)=L. \end{equation*}

This definition captures the ad hoc reasoning we used when looking at the graphs. We looked at x-values that were close to \(a\) but not equal and then looked at their corresponding y-values. Our eye told us if the y-values were getting closer and closer to some value \(L\) or not. Examining the graph like this is our first technique for trying to answer the question of whether or not a limit exists, and when it does, what the limit is equal to.

A second method is related but a bit more formal. The idea is to pick a sequence of numbers that approach \(a\) and evaluate them with our function. If we arrange these in a table, we can often see whether or not there is a limit.

Let's try this in an example. If we let

\begin{equation*} f(x)=\frac{\sin(x)}{x}, \end{equation*}

we see that it is undefined at \(x=0\text{,}\) yielding a zero in the numerator and denomentator. We set up a sequence of numbers that approach zero, using \(\{1,0.1,0.001,0.0001,0.00001,0.000001,0.0000001\}\) and evaluate them using the function. We can see the results in the two left-most columns of Table 1.5.3 and it looks like the values are converging to one. But what if we took negative numbers that approached zero? Well, the two right-most columns show the results using the negative versions of our list of numbers. Again, we see the values look like they are converging to one.

Table 1.5.3.
\(x\) \(\frac{\sin{x}}{x}\) \(x\) \(\frac{\sin{x}}{x}\)
1 0.841470985 -1 0.841470985
0.1 0.998334166 -0.1 0.998334166
0.01 0.999983333 -0.01 0.999983333
0.001 0.999999833 -0.001 0.999999833
0.0001 0.999999998 -0.0001 0.999999998
0.00001 1.000000000 -0.00001 1.000000000
0.000001 1.000000000 -0.000001 1.000000000
0.0000001 1.000000000 -0.0000001 1.000000000

It is important to emphasize that this does not show us what the limit is - we might have picked a poor set of numbers that approach zero or maybe the function's behavior changes radically between zero and 0.0000001. These two methods only give us a candidate for limits or a sense that the limit does not exist. But, we will continue to develop more precise methods in the coming videos.