For all three functions, the limit as x approaches 2 is 7. 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. Imagine what the table of input and output values would look like for and. 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. 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. I have a bit more groundwork to lay before you see why I include it.) 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.įunction 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. 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. TABLE 7-1 Input and Output Values of as x Approaches 2 Now, look at Table 7-1.įIGURE 7-1: The graphs of the functions of f, g, and h. 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). 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. Using three functions to illustrate the same limitĬonsider the function on the left in Figure 7-1.
It’s much easier to understand limits through examples than through this sort of mumbo jumbo, so take a look at some. 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. Got it? You’re kidding! Let me say it another way. More about those limits later in the chapter and in Chapter 8.) ( Note: This definition does not apply to limits where x approaches infinity or negative infinity. 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. Don’t worry if you don’t grasp the concept right away. In this chapter, I lay the groundwork for differentiation and integration by exploring limits and the closely related topic, continuity. 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 are fundamental for both differential and integral calculus. Limits and ContinuityĮvaluating functions with holes - break out the mothballs
#Definition of limit calculus examples plus#
Plus the plain English meaning: not lifting your pencil off the paper.Ĭalculating limits with your calculator. The mathematical mumbo jumbo about continuity . Limits, asymptotes, and infinity : Far out, man. 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. Calculus For Dummies, 2nd Edition (2014) Part III.
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