## Calculus For Dummies, 2nd Edition (2014)

### Part III. Limits

**IN THIS PART …**

Limits: The mathematical microscope that lets you sort of zoom in on a curve to the sub-, sub-, sub-atomic level, where it becomes straight.

*Limits*, *asymptotes*, and *infinity* : Far out, man.

The mathematical mumbo jumbo about *continuity* . Plus the plain English meaning: not lifting your pencil off the paper.

Calculating limits with algebra.

Calculating limits with your calculator.

### Chapter 7. Limits and Continuity

**IN THIS CHAPTER**

**Taking a look at limits**

**Evaluating functions with holes — break out the mothballs**

**Exploring continuity and discontinuity**

Limits are fundamental for both differential and integral calculus. The formal definition of a derivative involves a limit as does the definition of a definite integral. (If you’re a real go-getter and can’t wait to read the actual definitions, check out __Chapters 9__ and __13__.) Now, it turns out that after you learn the shortcuts for calculating derivatives and integrals, you won’t need to use the longer limit methods anymore. But understanding the mathematics of limits is nonetheless important because it forms the foundation upon which the vast architecture of calculus is built (okay, so I got a bit carried away). In this chapter, I lay the groundwork for differentiation and integration by exploring limits and the closely related topic, continuity.

*Take It to the Limit — NOT*

Limits can be tricky. Don’t worry if you don’t grasp the concept right away.

**Informal definition of** ** limit** (the formal definition is in a few pages): The limit of a function (if it exists) for some

*x*-value

*c*, is the height the function gets closer and closer to as

*x*gets closer and closer to

*c*from the left and the right. (

**This definition does not apply to limits where**

*Note:**x*approaches infinity or negative infinity. More about those limits later in the chapter and in

__Chapter 8__.)

Got it? You’re kidding! Let me say it another way. A function has a limit for a given *x*-value *c* if the function zeros in on some height as *x* gets closer and closer to the given value *c* from the left and the right. Did that help? I didn’t think so. It’s much easier to understand limits through examples than through this sort of mumbo jumbo, so take a look at some.

*Using three functions to illustrate the same limit*

Consider the function on the left in __Figure 7-1__. When we say that the limit of as *x* approaches 2 is 7, written as , we mean that as *x* gets closer and closer to 2 from the left and the right, gets closer and closer to a height of 7. By the way, as far as I know, the number 2 in this example doesn’t have a formal name, but I call it the *arrow-number*. The arrow-number gives you a horizontal location in the *x* direction. Don’t confuse it with the *answer* to the limit problem or simply the *limit,* both of which refer to a *y*-value or *height* of the function (7 in this example). Now, look at __Table 7-1__.

** FIGURE 7-1:** The graphs of the functions of

*f*,

*g*, and

*h.*

__TABLE 7-1__ Input and Output Values of as *x* Approaches 2

__Table 7-1__ shows that *y* is approaching 7 as *x* approaches 2 from both the left and the right, and thus the limit is 7. If you’re wondering what all the fuss is about — why not just plug the number 2 into *x* in and obtain the answer of 7 — I’m sure you’ve got a lot of company. In fact, if all functions were *continuous* (without gaps) like *f*, you *could* just plug in the arrow-number to get the answer, and this type of limit problem would basically be pointless. We need to use limits in calculus because of *discontinuous* functions like *g* and *h* that have holes.

Function *g* in the middle of __Figure 7-1__ is identical to *f* except for the hole at and the point at . Actually, this function, , would never come up in an ordinary calculus problem — I only use it to illustrate how limits work. (Keep reading. I have a bit more groundwork to lay before you see why I include it.)

The important functions for calculus are the functions like *h* on the right in __Figure 7-1__, which come up frequently in the study of derivatives. This third function is identical to except that the point has been plucked out, leaving a hole at and no other point where *x* equals 2.

Imagine what the table of input and output values would look like for and . Can you see that the values would be identical to the values in __Table 7-1__ for ? For both *g* and *h,* as *x* gets closer and closer to 2 from the left and the right, *y* gets closer and closer to a height of 7. For all three functions, the limit as *x* approaches 2 is 7.

This brings us to a critical point: When determining the limit of a function as *x* approaches, say, 2, the value of — or even whether exists at all — is totally irrelevant. Take a look at all three functions again where equals 7, is 5, and doesn’t exist (or, as mathematicians say, it’s *undefined*). But, again, those three results are irrelevant and don’t affect the answer to the limit problem.

**You don’t get to the limit.** In a limit problem, *x* gets closer and closer to the arrow-number *c*, but technically *never gets there,* and what happens to the function when *x* equals the arrow-number *c* has *no effect* on the answer to the limit problem (though for continuous functions like the function value equals the limit answer and it can thus be used to compute the limit answer)*.*

*Sidling up to one-sided limits*

One-sided limits work like regular, two-sided limits except that *x* approaches the arrow-number *c* from just the left or just the right. The most important purpose for such limits is that they’re used in the formal definition of a regular limit (see the next section on the formal definition of a limit).

To indicate a one-sided limit, you put a little superscript subtraction sign on the arrow-number when *x* approaches the arrow-number from the left or a superscript addition sign when *x* approaches the arrow-number from the right. Like this:

or

Look at __Figure 7-2__. The answer to the regular limit problem, , is that the limit does not exist because as *x* approaches 3 from the left *and* the right, is not zeroing in on the same height.

** FIGURE 7-2:** An illustration of two one-sided limits.

However, both one-sided limits do exist. As *x* approaches 3 from the left, zeros in on a height of 6, and when *x* approaches 3 from the right, zeros in on a height of 2. As with regular limits, the value of has no effect on the answer to either of these one-sided limit problems. Thus,

and

A function like in __Figure 7-2__ is called a *piecewise* function because it’s got separate pieces. Each part of a piecewise function has its own equation — like, for example, the following three-piece function:

Sometimes a chunk of a piecewise function connects with its neighboring chunk, in which case the function is continuous there. And sometimes, like with , a piece does not connect with the adjacent piece — this results in a discontinuity.

*The formal definition of a limit — just what you’ve been waiting for*

Now that you know about one-sided limits, I can give you the formal mathematical definition of a limit. Here goes:

**Formal definition of limit:** Let *f* be a function and let *c* be a real number.

exists if and only if

1. exists,

2. exists, and

3.

Calculus books always present this as a three-part test for the existence of a limit, but condition 3 is the only one you need to worry about because 1 and 2 are built into 3. You just have to remember that you can’t satisfy condition 3 if the left and right sides of the equation are both undefined or nonexistent; in other words, it is *not* true that *undefined = undefined* or that *nonexistent = nonexistent.* (I think this is why calc texts use the 3-part definition.) As long as you’ve got that straight, condition 3 is all you need to check.

**When we say a limit exists, it means that the limit equals a** *finite***number.** Some limits equal infinity or negative infinity, but you nevertheless say that they *do not exist.* That may seem strange, but take my word for it. (More about infinite limits in the next section.)

*Limits and vertical asymptotes*

A *rational* function like has vertical asymptotes at and . Remember asymptotes? They’re imaginary lines that the graph of a function gets closer and closer to as it goes up, down, left, or right toward infinity or negative infinity. is shown in __Figure 7-3__.

** FIGURE 7-3:** A typical rational function.

Consider the limit of the function in __Figure 7-3__ as *x* approaches 3. As *x* approaches 3 from the left, goes up to infinity, and as *x* approaches 3 from the right, goes down to negative infinity. Sometimes it’s informative to indicate this by writing,

and

But it’s also correct to say that both of these limits *do not exist* because infinity is not a real number. And if you’re asked to determine the regular, two-sided limit, , you have no choice but to say that it does not exist because the limits from the left and from the right are unequal.

*Limits and horizontal asymptotes*

Up till now, I’ve been looking at limits where *x* approaches a regular, finite number. But *x* can also approach infinity or negative infinity. Limits at infinity exist when a function has a horizontal asymptote. For example, the function in __Figure 7-3__ has a horizontal asymptote at , which the function gets closer and closer to as it goes toward infinity to the right and negative infinity to the left. (Going left, the function crosses the horizontal asymptote at and then gradually comes down toward the asymptote. Going right, the function stays below the asymptote and gradually rises up toward it.) The limits equal the height of the horizontal asymptote and are written as

and

You see more limits at infinity in __Chapter 8__.

*Calculating instantaneous speed with limits*

If you’ve been dozing up to now, WAKE UP! The following problem, which eventually turns out to be a limit problem, brings you to the threshold of real calculus. Say you and your calculus-loving cat are hanging out one day and you decide to drop a ball out of your second-story window. Here’s the formula that tells you how far the ball has dropped after a given number of seconds (ignoring air resistance):

·

· (where *h* is the height the ball has fallen, in feet, and *t* is the amount of time since the ball was dropped, in seconds)

If you plug 1 into *t, h* is 16; so the ball falls 16 feet during the first second. During the first 2 seconds, it falls a total of , or 64 feet, and so on. Now, what if you wanted to determine the ball’s speed exactly 1 second after you dropped it? You can start by whipping out this trusty ol’ formula:

Using the *rate,* or *speed* formula, you can easily figure out the ball’s average speed during the 2nd second of its fall. Because it dropped 16 feet after 1 second and a total of 64 feet after 2 seconds, it fell , or 48 feet from second to seconds. The following formula gives you the average speed:

But this isn’t the answer you want because the ball falls faster and faster as it drops, and you want to know its speed exactly 1 second after you drop it. The ball speeds up between 1 and 2 seconds, so this *average* speed of 48 feet per second during the 2nd second is certain to be faster than the ball’s *instantaneous* speed at the end of the 1st second. For a better approximation, calculate the average speed between second and seconds. After 1.5 seconds, the ball has fallen or 36 feet, so from to , it falls , or 20 feet. Its average speed is thus

If you continue this process for elapsed times of a quarter of a second, a tenth of a second, then a hundredth, a thousandth, and a ten-thousandth of a second, you arrive at the list of average speeds shown in __Table 7-2__.

__TABLE 7-2__ Average Speeds from 1 Second to *t* Seconds

As *t* gets closer and closer to 1 second, the average speeds appear to get closer and closer to 32 feet per second.

Here’s the formula we used to generate the numbers in __Table 7-2__. It gives you the average speed between 1 second and *t* seconds:

(In the line immediately above, recall that *t* cannot equal 1 because that would result in a zero in the denominator of the original equation. This restriction remains in effect even after you cancel the .)

__Figure 7-4__ shows the graph of this function.

__FIGURE 7-4:__*f(t)* is the average speed between 1 second and *t* seconds.

This graph is identical to the graph of the line except for the hole at . There’s a hole there because if you plug 1 into *t* in the average speed function, you get

which is undefined. And why did you get ? Because you’re trying to determine an average speed — which equals *total distance* divided by *elapsed time* — from to . But from to is, of course, *no* time, and “during” this point in time, the ball doesn’t travel any distance, so you get as the average speed from to .

Obviously, there’s a problem here. Hold on to your hat, you’ve arrived at one of the big “Ah ha!” moments in the development of differential calculus.

**Definition of** ** instantaneous speed:** Instantaneous speed is defined as the limit of the average speed as the elapsed time approaches zero.

For the falling-ball problem, you’d have

The fact that the elapsed time never gets to zero doesn’t affect the precision of the answer to this limit problem — the answer is exactly 32 feet per second*,* the height of the hole in __Figure 7-4__. What’s remarkable about limits is that they enable you to calculate the precise, instantaneous speed at a *single* point in time by taking the limit of a function that’s based on an *elapsed* time, a period between *two* points of time.

*Linking Limits and Continuity*

Before I expand on the material on limits from the earlier sections of this chapter, I want to introduce a related idea — *continuity.* This is such a simple concept. A *continuous* function is simply a function with no gaps — a function that you can draw without taking your pencil off the paper. Consider the four functions in __Figure 7-5__.

** FIGURE 7-5:** The graphs of the functions

*f*,

*g*,

*p*, and

*q*.

Whether or not a function is continuous is almost always obvious. The first two functions in __Figure 7-5__, and , have no gaps, so they’re continuous. The next two, and , have gaps at , so they’re not continuous. That’s all there is to it. Well, not quite. The two functions with gaps are not continuous everywhere, but because you can draw sections of them without taking your pencil off the paper, you can say that parts of those functions are continuous. And sometimes a function is continuous everywhere it’s defined. Such a function is described as being *continuous over its entire domain,* which means that its gap or gaps occur at *x*-values where the function is undefined. The function is continuous over its entire domain; , on the other hand, is not continuous over its entire domain because it’s not continuous at , which is in the function’s domain. Often, the important issue is whether a function is continuous at a particular *x*-value. It is unless there’s a gap there.

**Continuity of polynomial functions:** All polynomial functions are continuous everywhere.

**Continuity of rational functions:** All rational functions — a *rational function* is the quotient of two polynomial functions — are continuous over their entire domains. They are discontinuous at *x*-values not in their domains — that is, *x*-values where the denominator is zero.

*Continuity and limits usually go hand in hand*

Look at the four functions in __Figure 7-5__ where . Consider whether each function is continuous there and whether a limit exists at that *x*-value. The first two, *f* and *g*, have no gaps at , so they’re continuous there. Both functions also have limits at , and in both cases, the limit equals the height of the function at , because as *x* gets closer and closer to 3 from the left and the right, *y* gets closer and closer to and , respectively.

Functions *p* and *q,* on the other hand, are not continuous at (or you can say that they’re *discontinuous* there), and neither has a regular, two-sided limit at . For both functions, the gaps at not only break the continuity, but they also cause there to be no limits there because, as you move toward from the left and the right, you do not zero in on some single *y*-value.

So there you have it. If a function is continuous at an *x*-value, there must be a regular, two-sided limit for that *x*-value. And if there’s a discontinuity at an *x*-value, there’s no two-sided limit there … well, almost. Keep reading for the exception.

*The hole exception tells the whole story*

The hole exception is the only exception to the rule that continuity and limits go hand in hand, but it’s a *huge* exception. And, I have to admit, it’s a bit odd for me to say that continuity and limits *usually* go hand in hand and to talk about this *exception* because the exception is the whole point. When you come right down to it, the exception is more important than the rule. Consider the two functions in __Figure 7-6__.

** FIGURE 7-6:** The graphs of the functions

*r*and

*s*.

These functions have gaps at and are obviously not continuous there, but they *do* have limits as *x* approaches 2. In each case, the limit equals the height of the hole.

**The hole exception:** The only way a function can have a regular, two-sided limit where it is not continuous is where the discontinuity is an infinitesimal hole in the function.

So both functions in __Figure 7-6__ have the same limit as *x* approaches 2; the limit is 4, and the facts that and that is undefined are irrelevant. For both functions, as *x* zeros in on 2 from either side, the height of the function zeros in on the height of the hole — that’s the limit. This bears repeating, even an icon:

**The limit at a hole:** The limit at a hole is the height of the hole.

“That’s great,” you may be thinking. “But why should I care?” Well, stick with me for just a minute. In the falling ball example in the “__Calculating instantaneous speed with limits__” section earlier in this chapter, I tried to calculate the average speed during zero elapsed time. This gave me . Because is undefined, the result was a hole in the function. Function holes often come about from the impossibility of dividing zero by zero. It’s these functions where the limit process is critical, and such functions are at the heart of the meaning of a derivative, and derivatives are at the heart of differential calculus.

**The derivative-hole connection:** A derivative always involves the undefined fraction and always involves the limit of a function with a hole. (If you’re curious, all the limits in __Chapter 9__ — where the derivative is formally defined — are limits of functions with holes.)

*Sorting out the mathematical mumbo jumbo of continuity*

All you need to know to fully *understand* the idea of continuity is that a function is continuous at some particular *x*-value if there is no gap there. However, because you may be tested on the following formal definition, I suppose you’ll want to know it.

**Definition of continuity:** A function is *continuous* at a point if the following three conditions are satisfied:

1. is defined,

2. exists, and

3. .

Just like with the formal definition of a limit, the definition of continuity is always presented as a 3-part test, but condition 3 is the only one you really need to worry about because conditions 1 and 2 are built into 3. You must remember, however, that condition 3 is *not* satisfied when the left and right sides of the equation are both undefined or nonexistent.

*The 33333 Limit Mnemonic*

Here’s a great memory device that pulls a lot of information together in one swell foop. It may seem contrived or silly, but with mnemonic devices, contrived and silly work. The 33333 limit mnemonic helps you remember five groups of three things: two groups involving limits, two involving continuity, and one about derivatives. (I realize we haven’t gotten to derivatives yet, but this is the best place to present this mnemonic. Take my word for it — nothing’s perfect.)

First, note that the word *limit* has five letters and that there are five 3s in this mnemonic. Next, write *limit* with a lower case “l” and uncross the “t” so it becomes another “l” — like this:

**l i m i l**

Now, the two “l”s are for limits, the two “i”s are for continuity (notice that the letter “i” has a gap in it, thus it’s not continuous), and the “m” is for slope (remember ?), which is what derivatives are all about (you’ll see that in __Chapter 9__ in just a few pages).

Each of the five letters helps you remember three things — like this:

· 3 parts to the definition of a limit:

Look back to the definition of a limit in “__The formal definition of a limit — just what you’ve been waiting for__” section. Remembering that it has three parts helps you remember the parts — trust me.

· 3 cases where a limit fails to exist:

· At a vertical asymptote — called an infinite discontinuity — like at on function *p* in __Figure 7-5__.

· At a jump discontinuity, like where on function *q* in __Figure 7-5__.

· With a limit at infinity of an *oscillating function* like which goes up and down forever, never zeroing in on a single height.

· 3 parts to the definition of continuity:

Just as with the definition of a limit, remembering that the definition of continuity has 3 parts helps you remember the 3 parts (see the section “__Sorting out the mathematical mumbo jumbo of continuity__”).

· 3 types of discontinuity:

· A removable discontinuity — that’s a fancy term for a hole — like the holes in functions *r* and *s* in __Figure 7-6__.

· An infinite discontinuity like at on function *p* in __Figure 7-5__.

· A jump discontinuity like at on function *q* in __Figure 7-5__.

Note that the three types of discontinuity (hole, infinite, and jump) begin with three consecutive letters of the alphabet. Since they’re consecutive, there are no gaps between *h*, *i*, and *j*, so they’re continuous letters. Hey, was this book worth the price or what?

· 3 cases where a derivative fails to exist:

(I explain this in __Chapter 9__ — keep your shirt on.)

· At any type of *discontinuity*.

· At a sharp point on a function, namely, at a *cusp* or a *corner.*

· At a *vertical tangent* (because the slope is undefined there).

Well, there you have it. Did you notice that another way this mnemonic works is that it gives you 3 cases where a limit fails to exist, 3 cases where continuity fails to exist, and 3 cases where a derivative fails to exist? *Holy triple trio of nonexistence, Batman, that’s yet another 3 — the 3 topics of the mnemonic: limits, continuity, and derivatives!*