Both of your new functions must have the same limit as x. The basic idea is to find one function that’s always greater than the limit function (at least near the arrow-number) and another function that’s always less than the limit function. Figuring out volumes of odd shapes with the deli meat method Includes foundations in algebra, trigonometry, and pre-calculus concepts Explores sequences, series, and graphing common functions Instructs you how to.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.The sandwich or squeeze method is something you can try when you can’t solve a limit problem with algebra. Adding up the area between curves. One mean theorem: Random acts of kindness Don’t make me laugh. Phil: Use the Integral to Solve Problems.

In this chapter, I lay the groundwork for differentiation and integration by exploring limits and the closely related topic, continuity.Limits can be tricky. 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). (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. The formal definition of a derivative involves a limit as does the definition of a definite integral. Limits and ContinuityEvaluating functions with holes — break out the mothballsLimits are fundamental for both differential and integral calculus. Plus the plain English meaning: not lifting your pencil off the paper.Calculating limits with your calculator.

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. More about those limits later in the chapter and in Chapter 8.)Got it? You’re kidding! Let me say it another way. ( Note: This definition does not apply to limits where x approaches infinity or negative infinity.

The arrow-number gives you a horizontal location in the x direction. 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. 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. 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 limitConsider the function on the left in Figure 7-1.

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. Actually, this function, , would never come up in an ordinary calculus problem — I only use it to illustrate how limits work. 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. 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. 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. 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 2Table 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.

Take a look at all three functions again where equals 7, is 5, and doesn’t exist (or, as mathematicians say, it’s undefined). 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. 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.

Like this:Look at Figure 7-2. 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. 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. 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).

Thus,A function like in Figure 7-2 is called a piecewise function because it’s got separate pieces. As with regular limits, the value of has no effect on the answer to either of these one-sided limit problems. 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.

(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. 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. Here goes:Formal definition of limit: Let f be a function and let c be a real number.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. 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 forNow that you know about one-sided limits, I can give you the formal mathematical definition of a limit.

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. (More about infinite limits in the next section.)A rational function like has vertical asymptotes at and. That may seem strange, but take my word for it.

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. Limits at infinity exist when a function has a horizontal asymptote. But x can also approach infinity or negative infinity. 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.Up till now, I’ve been looking at limits where x approaches a regular, finite number. Sometimes it’s informative to indicate this by writing,But it’s also correct to say that both of these limits do not exist because infinity is not a real number. As x approaches 3 from the left, goes up to infinity, and as x approaches 3 from the right, goes down to negative infinity.

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. 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 asYou see more limits at infinity in Chapter 8.Calculating instantaneous speed with limitsIf 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.

How To Do Calculus For Dummies