APEX Calculus

We begin our study of limits by considering examples that demonstrate key concepts that will be explained as we progress.

Consider the function \(y = \frac{\sin(x) }{x}\text{.}\) When \(x\) is near the value 1, what value (if any) is \(y\) near?

While our question is not precisely formed (what constitutes “near the value 1”?), the answer does not seem difficult to find. One might think first to look at a graph of this function to approximate the appropriate \(y\) values. Consider Figure 1.1.3, where \(y = \frac{\sin(x) }{x}\) is graphed. For values of \(x\) near 1, it seems that \(y\) takes on values near \(0.85\text{.}\) In fact, when \(x=1\text{,}\) then \(y=\frac{\sin(1) }{1} \approx 0.84\text{,}\) so it makes sense that when \(x\) is “near” 1, \(y\) will be “near” \(0.84\text{.}\)

Consider this same function again at a different value for \(x\text{.}\) When \(x\) is near \(0\text{,}\) what value (if any) is \(y\) near? By considering Figure 1.1.4, one can see that it seems that \(y\) takes on values near \(1\text{.}\) But what happens when \(x=0\text{?}\) We have

The expression \(0/0\) has no value; it is indeterminate. Such an expression gives no information about what is going on with the function nearby. We cannot find out how \(y\) behaves near \(x=0\) for this function simply by letting \(x=0\text{.}\)

Finding a limit entails understanding how a function behaves near a particular value of \(x\text{.}\) Before continuing, it will be useful to establish some notation. Let \(y=f(x)\text{;}\) that is, let \(y\) be a function of \(x\) for some function \(f\text{.}\) The expression “the limit of \(y\) as \(x\) approaches \(1\)” describes a number, often referred to as \(L\text{,}\) that \(y\) nears as \(x\) nears \(1\text{.}\) We write all this as

This is not a complete definition (that will come in the next section); this is a pseudo-definition that will allow us to explore the idea of a limit.

Above, where \(f(x) = \sin(x)/x\text{,}\) we approximated

(We approximated these limits, hence used the “\(\approx\)” symbol, since we are working with the pseudo-definition of a limit, not the actual definition.)

Once we have the true definition of a limit, we will find limits analytically; that is, exactly using a variety of mathematical tools. For now, we will approximate limits both graphically and numerically. Graphing a function can provide a good approximation, though often not very precise. Numerical methods can provide a more accurate approximation. We have already approximated limits graphically, so we now turn our attention to numerical approximations.

Consider again \(\lim_{x\to 1}\frac{\sin(x)}{x}\text{.}\) To approximate this limit numerically, we can create a table of \(x\) and \(f(x)\) values where \(x\) is “near” \(1\text{.}\) This is done in Figure 1.1.6.

Notice that for values of \(x\) near \(1\text{,}\) we have \(\sin(x)/x\) near \(0.841\text{.}\) The \(x=1\) row is included, but we stress the fact that when considering limits, we are not concerned with the value of the function at that particular \(x\) value; we are only concerned with the values of the function when \(x\) is near 1.

Now approximate \(\lim_{x\to 0} \frac{\sin(x)}{x}\) numerically. We already approximated the value of this limit as \(1\) graphically in Figure 1.1.4. Figure 1.1.7 shows the value of \(\sin(x)/x\) for values of \(x\) near \(0\text{.}\) Ten places after the decimal point are shown to highlight how close to \(1\) the value of \(\sin(x)/x\) gets as \(x\) takes on values very near \(0\text{.}\) We include the \(x=0\) row but again stress that we are not concerned with the value of our function at \(x=0\text{,}\) only on the behavior of the function near \(0\text{.}\)

This numerical method gives confidence to say that \(1\) is a good approximation of \(\lim_{x\to 0} \frac{\sin(x)}{x}\text{;}\) that is,

Later we will be able to prove that the limit is exactly \(1\text{.}\)

We now consider several examples that allow us explore different aspects of the limit concept.

Example 1.1.8. Approximating the value of a limit.

Use graphical and numerical methods to approximate

\begin{equation*} \lim_{x\to 3} \frac{x^2-x-6}{6x^2-19x+3}\text{.} \end{equation*}

Solution

To graphically approximate the limit, graph

\begin{equation*} y = \frac{x^2-x-6}{6x^2-19x+3} \end{equation*}

on a small interval that contains \(3\text{.}\) To numerically approximate the limit, create a table of values where the \(x\) values are near \(3\text{.}\) This is done in Figure 1.1.9 and Figure 1.1.10, respectively.

\(x\)	\(\frac{x^2-x-6}{6x^2-19x+3}\)
\(2.9\)	\(0.29878\)
\(2.99\)	\(0.294569\)
\(2.999\)	\(0.294163\)
3	not defined
\(3.001\)	\(0.294073\)
\(3.01\)	\(0.293669\)
\(3.1\)	\(0.289773\)

The graph shows that when \(x\) is near \(3\text{,}\) the value of \(y\) is very near \(0.3\text{.}\) By considering values of \(x\) near \(3\text{,}\) we see that \(y=0.294\) is a better approximation. The graph and the table imply that

\begin{equation*} \lim_{x\to 3} \frac{x^2-x-6}{6x^2-19x+3} \approx 0.294\text{.} \end{equation*}

This example may bring up a few questions about approximating limits (and the nature of limits themselves).

1. If a graph does not produce as good an approximation as a table, why bother with it?

2. How many values of \(x\) in a table are “enough?” In the previous example, could we have just used \(x=3.001\) and found a fine approximation?

Graphs are useful since they give a visual understanding concerning the behavior of a function. Sometimes a function may act “erratically” near certain \(x\) values which is hard to discern numerically but very plain graphically (see Example 1.1.22). Since graphing utilities are very accessible, it makes sense to make proper use of them.

Since tables and graphs are used only to approximate the value of a limit, there is not a firm answer to how many data points are “enough.” Include enough so that a trend is clear, and use values (when possible) both less than and greater than the value in question. In Example 1.1.8, we used both values less than and greater than \(3\text{.}\) Had we used just \(x=3.001\text{,}\) we might have been tempted to conclude that the limit had a value of \(0.3\text{.}\) While this is not far off, we could do better. Using values “on both sides of 3” helps us identify trends.

Example 1.1.11. Approximating the value of a limit.

Graphically and numerically approximate the limit of \(f(x)\) as \(x\) approaches \(0\text{,}\) where

\begin{equation*} f(x) = \begin{cases}x+1 \amp x\lt 0 \\ -x^2+1 \amp x \gt 0\end{cases}\text{.} \end{equation*}

Solution

Again we graph \(f(x)\) and create a table of its values near \(x=0\) to approximate the limit. Note that this is a piecewise defined function, so it behaves differently on either side of \(0\text{.}\) Figure 1.1.12 shows a graph of \(f(x)\text{,}\) and on either side of \(0\) it seems the \(y\) values approach \(1\text{.}\) Note that \(f(0)\) is not actually defined, as indicated in the graph with the open circle.

\(x\)	\(f(x)\)
\(-0.1\)	\(0.9\)
\(-0.01\)	\(0.99\)
\(-0.001\)	\(0.999\)
\(0.001\)	\(0.999999\)
\(0.01\)	\(0.9999\)
\(0.1\)	\(0.99\)

Figure 1.1.13 shows values of \(f(x)\) for values of \(x\) near \(0\text{.}\) It is clear that as \(x\) takes on values very near \(0\text{,}\) \(f(x)\) takes on values very near \(1\text{.}\) It turns out that if we let \(x=0\) for either “piece” of \(f(x)\text{,}\) \(1\) is returned; this is significant and we'll return to this idea later.

The graph and table allow us to say that \(\lim_{x\to 0}f(x) \approx 1\text{;}\) in fact, we are probably very sure it equals 1.