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3.1: Functions, Limits, and Continuity

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Equations and Graphs

Numbers or shapes. Which is more basic? Numbers can be used to describe relationships among shapes, but shapes can also be used to represent numbers and equations. So are numbers based upon shapes or are shapes just representations of numbers?

Throughout much of our mathematical history numbers and geometry have remained two more or less separate fields, difficult to reconcile in any universal way. People who studied shapes, like Euclid and Archimedes, would attach equations and numbers to their figures but without any precise meanings. And early number theorists and algebraists like Fibonacci and Fermat would use pictures to understand their equations but also not in any methodical way.

Finally though, René Descartes (1596 – 1650) discovered a sensible and agreed-upon system for connecting geometry and analysis. The concept of Cartesian Coordinates, or using a horizontal distance and a vertical distance to give a point’s location in the plane, changed the course of human thinking. Of course, he didn’t call them Cartesian Coordinates since that would have been a little self-indulgent. But in any case, this work laid the foundation for analytic geometry by making a framework in which equations and functions could finally be represented by shapes, and shapes could be discussed in terms of algebraic formulas.

The simplest graphs display no more than a list of points. For example:

(0, 1) \ \ (1, 3) \ \ (3, 4) \ \ (-2, 7) \ \ (-1, -1) \ \ (2, -1)

Equations, on the other hand, can be thought of as a huge (infinite actually) list of points corresponding to every pair (x , y) that solves the equation. However, the two are not the same. For example, some lists of coordinate-pairs have no corresponding equation (like the list above), and some have an infinite number of different possibilities (like any list which does have an equation for example). Similarly, a given equation may have no corresponding points as solutions (like y^2 = -1), may be solved by any point in the plane (like \frac{y^2-x^2}{y+x} = y - x), or may have a curve as its set of solution points like the examples in this lesson.

Relations and Functions

Historically, the term of function is due to Gottfried Leibniz who along with Newton is credited with discovering calculus. He used the term to describe properties of a curve as one moves along it, such as its length, height above the x-axis, or steepness.

However, the concept of functions is essential to all areas of math and has a nice generalization. In full generality a function maps one set or collection of objects into another set. Consider the two sets below

We can think of various functions that relate these two, such as the one that assigns to each person his/her age. Since a person cannot simultaneously be two different ages, we require that all objects in A are mapped to exactly one object in B. This is a basic requirement of functions. However, various people may have the same age so it may be that one object in B is the image of various objects in A.

Functions can also be understood geometrically. This way of thinking allows another understanding of the requirement that each object in the domain correspond to a unique object in the range. The idea is that one set should be crumpled up if necessary, and then physically placed on top of another. If the set A of people is placed on top of or inside the set B of numbers, each object A sits over just one number, not two different numbers. Now, the set A may be twisted and bent so that two people are lying on top of the same number. But a single person can only be in one place.

A parabola, for example, can be thought of by taking two copies of the real number line as the sets A and B. Then fold the number line A across zero and place this over the positive half of B so that zero meets zero, one meets one, two meets four, three meets nine, and so on. Similarly, the sine function might be better understood by taking the same sets A and B of real numbers. Then fold A at each integer multiple of 2 \pi back over itself and lie this along the interval of B between 1 and -1.

Finally, suppose A is mapped into B by a function f and B into C by a function g. Then the composition f  \ o \ g can be represented by first laying the set B over C so that elements of B lie on top of their image in C and then placing A on top of the crumpled up version of B in C.

Models and Data

Data can be given in various forms. For example, suppose you are designing a sky-scraper and need to know the temperature and pressure at different heights above the ground. Then the data may look something like:

altitude temperature pressure
0\;\mathrm{ft} 85\;\mathrm{F} 1.000\;\mathrm{atm}
10\;\mathrm{ft} 84\;\mathrm{F} 0.999\;\mathrm{atm}
20\;\mathrm{ft} 83\;\mathrm{F} 0.998\;\mathrm{atm}
30\;\mathrm{ft} 82\;\mathrm{F} 0.995\;\mathrm{atm}
40\;\mathrm{ft} 81\;\mathrm{F} 0.990\;\mathrm{atm}
50\;\mathrm{ft} 80\;\mathrm{F} 0.980\;\mathrm{atm}

This data suggests two functions: one giving the temperature T as a function of altitude y and one giving the pressure P as a function of height y. Scientists, engineers, and many other people must fit their data to functions all the time, and in fact, when this is done well it can produce incredible results. Numerous Nobel prizes in physics and other sciences are the result of careful measurements and data fitting.

One of the most noteworthy examples of how data fitting can lead to great advances is that of Max Plank and the so-called blackbody radiation problem. People have known for many millennia that when objects are heated they give off light. However, every model created to fit data before Plank came along predicted something that was not observed: The so-called ultraviolet catastrophe. However, Plank looked at the same data and used a little creativity in fitting this to a model and ended up changing physics in a drastic way forever. For this work he was given the 1918 Nobel Prize in physics!

The most basic data modeling problem is that one is given a finite data set containing, say, n points of the form (x_i,y_i) for i = 1, \ldots , n. Then there is a famous theorem that this data can be fit exactly by a polynomial of any order greater than or equal to n-1. That is, a single point can be fit with a line, or a parabola, or anything. Two points can be fit with a line, a parabola or anything as well. But, while it may or may not be possible to fit three points with a line, it is guaranteed that they can be fit by a parabola or any higher order polynomial. Similarly, four arbitrary points may not be able to be perfectly modeled by a line or a parabola, but can be fit along a cubic or higher order polynomial. The theorem has many names and is attributed to various people since it was independently proven at various times in history.

In terms of this theorem, trigonometric and exponential fits are appealing since the exponential and trigonometric functions are like polynomials with an infinite number of terms. One thus expects that using the correct coefficients, an arbitrary number of points could be fit to a combination of exponential or trig functions. In fact, this is the basis of Fourier analysis!

The Calculus

The concept of slope is likely to be familiar. But the idea of finding slope for a curve that is not straight may be less so. It was nearly 1700 before Issac Newton and Gottfried Leibniz figured out how to do this. Each realized that a curve has a different steepness at each point, and came up with an ingenious way for calculating these. In essence, the idea was that secant lines become closer and closer to a special line (called the tangent) when two points approach one-another. This is a very geometric concept, and it is probably best introduced as such:

Newton and Leibniz also investigated areas in the 18^{\mathrm{th}} century and recognized the relationship contained in the Fundamental Theorem of Calculus. But it wasn’t until the 19^{\mathrm{th}} century and the work of Bernhard Riemann that integration was formalized using the limits suggested in this chapter.

The connection between differentiation and integration is, again, a geometric one. This can be thought of without limits and rigor and therefore could enrich this chapter. Let f(x) be a curve and suppose we define F(z) to be the area under this curve between the origin and an arbitrary point x. Then the change in F(z) at the point z is the rate at which area is increasing as we move through x. But this is just given precisely by the height f(z):

Finding Limits

One might say that all of math is essentially an attempt to understand infinity. And this is nowhere more evident than in our attempt to give concrete meaning to concepts like limits and continuity. Calculus cannot be done without carefully considering how a function behaves as one makes the argument closer and closer to a certain value. However this concept was only defined in the epsilon-delta or open set sense during the 19^{\mathrm{th}} century, after calculus had already been developed by Newton and Leibniz. Bernard Bolzano and Augustin-Louis Cauchy deserve most of the credit for this (unfortunately complicated) definition of limits.

While it is true that the formal definition for a limit’s existence is notably awkward, this is especially the case in the form presented here. The points a and L are enclosed in “open intervals,” however; students are not likely to be familiar with this terminology. What is an open interval instead of a closed one? And indeed, do you really expect me to keep all these letters straight: x, \ a, \ L, \ D?

This definition is better introduced by discussing in detail how and why it fails. For example, consider the piecewise function defined by:

f(x) = \begin{Bmatrix}  x + 1 & x \le 0 \\  x - 1 & x > 0\end{Bmatrix}

And suppose we’re interested in finding out if the following limit exists:

\lim_{x \to 0} f(x) = L

We can choose any range along the y-axis in order to zoom in more closely on what’s going on here at the origin. If we let y range from -100 to +100, we don’t really see anything strange at x = 0.

However, if we choose a smaller range for y we can see what is happening here:

The point is that if we zoom in too closely by narrowing the range of y-values, we will eventually find that the function jumps suddenly from -1 to +1. For any y-value less than -1, we can find an x that gives that value. And for any y-value greater than +1 we can similarly find an x giving that value. However there are two problems:

i) Is the limit -1 or +1?

ii) What about for y-values between -1 and +1? What x gives these values?

It can in fact be said that anytime a limit fails to exist, there is some kind of jump in the function like this.

Evaluating Limits

It is probably best to draw lots of graphs when teaching this lesson. The idea of a limit is really a geometric one and if it is introduced without pictures students are likely to be lost. The basic idea is that any function you can write down easily that is defined for some x-value a, will satisfy:

\lim_{x \to a}f(x) = f(a)

The “properties” should be introduced by simply saying that usually you can just let your instincts guide you.

In fact, it may make more sense to teach the lesson EVALUATING LIMITS before the one titled FINDING LIMITS since it is important to build intuition before making concrete definitions. The section on composite functions should be introduced a little by discussing composite functions. For example, you could start with some examples by explaining that sometimes complicated functions, like f(x)= \sqrt{1-x^2}, can be understood by thinking of them as two composed functions.

Example 5 should be clarified to students. While it is true that plugging in gives an indeterminate form, this does not mean anything about the limit. It just means we are going to work a little bit harder to find it, if it does exist. Now, in this case the limit really doesn’t exist. But we cannot see this simply by plugging in.

The fluctuation between very advanced descriptions and extremely basic ones in this chapter is regrettable. It is probably best to describe the squeeze theorem qualitatively and leave out the example unless students are really prepared. Otherwise the difficulty involved with understanding how to apply this theorem will surely lose most students.

Continuity Finding Limits

Continuity is a subject with considerable history and a very simple geometric idea at its heart. Basically, a continuous function is one that behaves as expected based on nearby points. This is why the definition is very simply that f(x) is continuous at x = a if:

\lim_{x \to a} f(x) = f(a)

For this statement to make sense, of course, we need that f(a) is defined and that the limit in question exists. However, it is best for students to get the idea of this definition. Later, they can be made to understand that requirements 1 and 2 are simply there to make sure that the formula above makes sense.

Really the idea is very simple: If you want to decide whether or not a function is continuous at some point a, just draw the function. If you have to lift your pen/cil off the page as you pass the point a then it is not continuous, otherwise it is. It really is that simple, and all of the formality should be introduced as a means towards making this concept concrete.

Notice that in the following graphs we cannot pass the value a = 3 without lifting the pencil

whereas in the following we can:

Infinite Limits

Here we confront head-on the concept of infinity, as is necessary when talking concretely about limits. The symbol \infty represents the positive tail of numbers as they get larger and larger, however \infty itself is not a number. For this reason, “equations” like

\infty + 2 & = \infty\\\frac{1}{\infty} & = 0\\\frac{1}{0} & = \infty

actually have no meaning. They may make sense in some formal sense where you mentally replace the symbol \infty by some huge number and the number 0 by a very small positive number. However this is not really concrete.

When dealing with infinity it is important to understand that no matter how big of a number you can imagine; infinity is still infinitely bigger. However, it is also important to be able to mentally plug in big numbers in order to obtain ideas of what will happen as we approach infinity. For example, consider the limit:

\lim_{x \to \infty} \frac{3x^3 - 4x^2 + 5}{9x^7 - x^3 + x^2 - 5x + 2}

As we plug in larger and larger numbers, it should be clear that the denominator will get much bigger than the numerator. This means the overall fraction will get smaller and smaller so we may guess it approaches zero. And in fact this is the case, since dividing top and bottom by x^7 we see the numerator goes to zero and the denominator goes to 9.

What on the other hand might we guess for the function:

\lim_{x \to \infty} \frac{3x^7 - 4x^2 + 5}{9x^7 - x^3 + x^2 - 5x + 2}

Here it is not clear what happens for large x since both the numerator and denominator will be big. However the same trick of dividing top and bottom by x^7 will make it clear that the limit should be

\frac{3}{9} = \frac{1}{3}

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