Finding Limits
Finding limits in mathematics involves determining the value that a function approaches as the input approaches a certain value. This concept is essential for understanding the behavior of functions near specific points and is a fundamental aspect of calculus. By finding limits, mathematicians can analyze the behavior of functions and solve various problems in calculus and other areas of mathematics.
4 Key excerpts on "Finding Limits"
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...In the figure below, the shaded area between the graph of f (x), the lines x = –2, x = 1, and the x -axis is 26.25. Chapters 7 and 8 are concerned with the issue of the area problem and the doors that this problem opens. DID YOU KNOW? Sir Isaac Newton (1642–1727) actually discovered calculus between 1665 and 1667 after his university closed due to an outbreak of the plague. Newton was only 22 at the time, and he preferred not to publish his discoveries. Meanwhile, in Germany, Gottfried Wilhelm Leibniz discovered calculus independently and was very open with his findings. This led to a bitter dispute between the two mathematicians later known as the “Great Sulk.” Today it is well known that both men discovered calculus independently of the other, Leibniz about 8 years after Newton. Graphical Approach to Limits Overview: The backbone of calculus is limits. While only about 5% of the AP exam actually tests limits directly, all of the concepts of differential and integral concept are based on limits. In this section, we will find limits when we are given graphs of functions. A limit is a boundary. A speed limit is the maximum speed you can travel without being in danger of receiving a ticket. A student who knows his limits with his parents knows exactly how far he can push his parents without them coming down on him. In real life, sometimes a limit can be somewhat hazy. For instance, I might not be sure of my limit of tolerance of heat before I turn the air conditioner on. But I do know that 70 degrees is on one side of that limit and 90 degrees is on the other side. The study of limits is crucial to calculus, dictating that most calculus courses begin with a study of them. The notation for limits is typically: What this says is: as the value of x gets close to the value of c, the value of y gets close to some constant L. The figure above demonstrates this: The closer the x -value is to c, the closer the y -value is to L. Whether the y -value ever reaches L is irrelevant...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 2 Limits and Continuity CHAPTER 2 LIMITS AND CONTINUITY 2.1 INTRODUCTION Limits are the building blocks of calculus. They are used to establish all major concepts, including continuity, derivatives, and integrals. There are many ways to examine and evaluate limits. Some of the methods that we will review are studying numerical patterns, direct substitution, deducing information from a graph, simplifying prior to substitution, and taking an intuitive approach to the behavior of a particular function. Limits are also closely linked with continuity. Prior to establishing a formal definition of continuity, you should recall from previous courses that any kind of a break in the domain of a function is called a discontinuity. Whether a function is continuous or discontinuous at a point in its domain determines the ease with which a limit at that point may be determined. Figure 2.1 Imagine a square with an area of 4 square feet (Figure 2.1). If the consecutive midpoints of each side of the square are connected with segments, a new square is formed with an area of 2 square feet. If the consecutive midpoints of the new square are connected, the resulting area is 1 square foot. Repeating this pattern over and over, the sequence of areas is 4, Even though each new area could continue to be multiplied by forever, it can be said that the limit of the areas is 0. Interestingly, you may recognize this sequence from a previous course as an infinite geometric sequence with a common ratio of The sum of the infinite number of consecutive areas has a limit given by the formula where a is the first term, and r is the common ratio. In this case, square feet. If n is the number of squares whose areas are being summed, the previous conclusion could also be written as 2.2 LIMITS AS X APPROACHES A CONSTANT A NUMERICAL APPROACH Table 2.1 Table 2.1 lists ordered pairs for a certain function, f (x)...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...4 Limits and derivatives In mathematics, differential calculus is a subfield of calculus that is concerned with the study of how quickly functions change over time. The primary concept in differential calculus is the derivative function. The derivative allows us to find the rate of change of economic variables over time. This chapter introduces the concept of a derivative and lays out the most important rules of differentiation. To properly introduce derivatives, one needs to consider the idea of a limit. We cover the concept of a limit in the first section. The chapter closes with growth rates of discrete and continuous variables. 4.1 Limits Consider a function g given by and shown in Figure 4.1. Clearly, the function is undefined for x = 0, since anything divided by zero is undefined. However, we can still ask what happens to g (x) when x is slightly above or below zero. Using a calculator we can find the values of g (x) in the neighborhood of x = 0, as shown in Table 4.1. As x approaches zero, g (x) takes values closer and closer to 2. So we can say that g(x) tends to 2 as x tends to zero. We write and say that the limit of g (x) as x approaches zero is equal to 2. Now that the idea of a limit is clear on an intuitive level, let us consider a formal definition of the right- and left-hand side limits. Let f be a function defined on some open interval (a, b). We say that L is the right-hand side limit of f (x) as x approaches a from the right and write if for every ε > 0 there is a δ > 0 such that Figure 4.1 Table 4.1 whenever As an example, let us consider the following function We want to show that Let us choose ε > 0. We need to show that there is a δ > 0 such that whenever Let us choose δ = (ε/ 2). Then, and therefore It follows immediately that whenever Now we have proved that the limit of as x approaches zero from the right is equal to 1. Now let us define a left-hand side limit. Let f be a function defined on some open interval (a, b)...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...The graph shifts to the right a units when a is subtracted from x. When graphing a function on the calculator (TI-83 or TI-84), make sure that all the plots are turned off; otherwise you risk getting an error and not being able to graph. To turn off the plots, press and place the cursor on the plot you want to deactivate (whichever is highlighted). Press. An even-degree polynomial with a positive leading coefficient has y -values that approach infinity as x → ±∞ (both ends go up). If the polynomial has a negative leading coefficient, its y -values approach negative infinity as x → ±∞ (both ends go down). An odd-degree polynomial with a positive leading coefficient has y -values that approach infinity as x → ∞ and y -values that approach negative infinity as x → –∞ (the right end goes up and the left end goes down). If the polynomial has a negative leading coefficient its y -values approach negative infinity as x → ∞; as x → –∞ its y -values approach positive infinity (the right end goes down and the left end goes up). CHAPTER 2 PRACTICE PROBLEMS (See solutions on page 195) For each of the functions in problems 1–8, draw the mother function and the given function on the same set of axes. 2. y = 2 |3 x + 4| 5. y = e x +2 – 1 6. y = ln(4 – x) Chapter 3 Limits of Functions I. MEANING OF LIMIT A. The limit of a function, y = f (x), as x approaches a number or ±∞, represents the value that y approaches. B. The left-hand limit,, states that. as x approaches a, from the left of a, f (x) approaches L. The right-hand limit,, states that as x approaches a, from the right of a, f (x) approaches L. C. The expression states that as x approaches a, simultaneously from the left and right of a, f (x) approaches L. D. The limit of a function at a point exists if and only if the left- and right-hand limits exist and are equal. Symbolically, if and then...
