This document discusses limits involving infinity. It defines an infinite limit as one where the function tends to positive or negative infinity as the variable approaches a value. An infinite limit is written as lim f(x) = +∞ if it tends to positive infinity, and lim f(x) = -∞ if it tends to negative infinity. As an example, the limit of the function f(x) = 1/(x-4)2 as x approaches 4 is written as lim f(x) = ∞, since the function grows without stopping as x gets closer to 4.
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)SNSDTaeyeon
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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)SNSDTaeyeon
This is my first upload in slideshare. I hope you guys like it~! and... Note: My fonts used are the ff:
1. exoziti.zip;
2. exoplanet.zip;
3. vlaanderen.zip;
4.Girls Generation Fonts.zip; and,
5. kimberly-geswein_over-the-rainbow.zip...
I hope you guys like it~!
add me on fb: www.fb.com/iamsieghart
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This is meant for university students taking either information technology or engineering courses, this course of differentiation, Integration and limits helps you to develop your problem solving skills and other benefits that come along with it.
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The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Unit 8 - Information and Communication Technology (Paper I).pdf
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1. 2.6 THEOREMS OF LIMITS Given the functions f(x), g(x), the value “a” (the one that x approaches), and its limits: Lím f(x) = L lím g(x) = M xaxa 1. For a constant f(x) = k Lím k = k xa Example: Lím 3 = 3 x2 2. For an independent variable f(x) = x Límx = a xa Example: lím x=3 x3
2. For a power f(x) = xn, with n = integer Lim xn = an xa Example: Lím x2 = (3)2 = 9 x3 For a constant by a power f(x) = kxn, with n = integer Lim kxn = k (lim xn) = k(an) Example: Lim kxn = k(lim xn) = k(an) xaxa Example: Lim 3x2 = 3 (lim x2) = 3(42) = 3 (16) = 48
3. For a sum of functions Lim [f(x) + g(x)] = lim f(x) + lim g(x) = L+M xaxaxa example: lim [3x2 + 6x] = lim 3x2 + lim 6x = 12+ 12 = 24 x2 x2x2 For a product of functions Lim [f(x)g(x) = [lim f(x)][lim g(x)] = (L)(M) xaxaxa Example: lim [(3x2)(6x)] = [lim 3x2][lim 6x] = (12)(12) = 144 x2 x2x2 For a quotient of functions Lim [f(x) ÷g(x)] = [lim f(x)]÷[lim g(x)] = (L) ÷ (M), if M0 xaxaxa Example: lim [(3x2)2÷(6x)] = [lim 3x2]÷[lim 6x] = (12) ÷ (12) = 1 x2 x2x2
4. For a function elevated to a power Lim [f(x)n = [lim f(x)]n = (L)n, with n = integer Example: Lim (3x2)2 = (lim 3x2)2 = [(3)(lim x2)2 = [(3)(22)2 = [12]2 = 144 x2 x2x2 Lim [3x2 + 6x]2 = [lim 3x2 +6x]2 = [lim 3x2 + 6x]2 = [3 lim x2 + 6lim x]2 = [3(2)2 + 6(2)]2 = [12+12]2 = [24]2 = 576 For a root of function Lim nf(x) = lim f(x) = L, with n= integer, f(x) ≥0 xaxa Example: lim4x2 = lim 4x2 = 4(2)2 = 16 = 4 x2 x2
12. 2.7 LATERAL LIMIT It is stated that the limit exist if the lateral limits are equals. The lateral limit is the value to which the function approaches when the variable “x” approaches to a value “a” either by its left side or by its right side. The following figures show the value “L” to which the function approaches when “x” approaches to a “a” either by a left of by the right.
14. Notice how when the variable “x” approaches to “a”, the function approaches to “L”, and the graph shows that its points are approaching more each time to the position (a, L) The lateral limits are two: the lateral limit by the left and the lateral limit by the right. The lateral limit by the left is the value to which approaches the function when x approaches to a by the left. It is written: lim f(x) = L1 xa- The lateral limit by the right is the value to which approaches the function when x approaches to a by the right. It is written: lim f(x) = L2 xa+
15. The lateral limits, L1 and L2, are equal, then the limit of the function when x approaches to “a”, exists; instead, if they are different, it is said that the limit does not exists. If lim f(x) = lim f(x) = L, then lim f(x) =L xa-xa+ if lim f(x) ≠ lim f(x), then lim f(x) = does not exists xa- xa+ Example 1 The following figure is the graph of f(x) = 0.1x3 – 0.5 x2 + 0.5x + 3.3. Observe the values to which the function approaches when the variable “x” approaches to 3. Notice that the lateral limits tend to a same value “L”.
16. You can see that when x approaches to 3 by the left, the limit is 3, and when x approaches to 3 by the right is also 3. Since when the variable “x” tends to 3, by both sides, the function tends to 3, then the limit is the function when x tend to 3 is 3.
17. EXAMPLE 2 The following figure is the graph of the function in parts, f(x) = x2 -4x + 5, if x < 3 - 0.5x + 5, if x ≥3
18. Observe the values to which the function approaches when the variable “x” approaches to 3. Notice that the lateral limits tend to be different values. You can see that when x approaches to 3 by the left, the limit is 2, and when x approaches to 3 by the right the limit is 3.5. Since when the variable “x” tends to different values by both sides, then the limit of the function when x tends to 3, does not exists.
19. EXAMPLE 3 Determine the following limit: lim 1 x4 x-4 Solution: Two tables of values are elaborated; in one table it is given values to “x” that get closer to 4 by the left and in the other table, with values of “x” that get closer to 4 by the right. When solving the corresponding values of f(x) you will observe the value to which the function approaches. 3 4 5
21. As the variable x approaches to 4, the function tends to different values. In the first table it is observed that when x approaches to 4 by the left, the values of the function f(x) tends to -∞, and in the second table you see that when x approaches to 4 by the right, the values of the function f(x) tend to +∞. Therefore, lim 1 = does not exist x4 x-4
25. 19. lim x-2 = x2+ 20. lim x-2 = x2- 21. f(x) = -x2 if x≤0, a) lim f(x) = b) lim f(x) = lim f(x) = x+1 if x>0 x0- x0+ x0 22. f(x)= x2-1 if x ≤1, a) lim f(x) = b) lim f(x) = lim f(x)= -x+3 if x>1 x1- x1+ x1
26. 23. F(x)= x2+3 if x≥1, a) lim f(x) = b) lim f(x)= lim f(x)= -x+3 if x>1 x1- x1+ x1 24. F(x)= x2-1 if x<1, a) lim f(x) = b) lim f(x)= lim f(x)= 1-x if x≥1 x1- x1+ x1 25. F(x)= 2 if x≤1, a) lim f(x) = b) lim f(x)= lim f(x)= -2 if x>1 x1- x1+ x1
27. Determine the indicated limit of the following graphs 30. Graph of the function f(x) = x2- 4x + 5 Lim f(x) = b) lim f(x)= c) lim f(x)= d) lim f(x)= e) lim f(x)= x0 x1 x2- + x x4
28. 31. Graph of the function f(x)= x2-2x+1 if<2 X2-6x+10 if x≥2 a)lim f(x) = b) lim f(x) = c) lim f(x) = d) lim f(x) = e) lim f(x) = x0 x1 x 2- x 2+ x2
29. 32. Graph of the function f(x) = 0.5/(x-3)2. a)lim f(x) = b) lim f(x) = c) lim f(x) = d) lim f(x) = e) lim f(x) = x2 x2.5 x 3- x 3+ x4
30. 33. Based on the following figure, find the limit for each case. a)lim f(x) = b) lim f(x) = c) lim f(x) = d) lim f(x) = e) lim f(x) = x0 x4x 3+ x 3- x5
31. 2.8 LIMITS WHERE IS INVOLVED THE INFINITE 2.8.1 Infinite Limit The infinite limit is the one where the function tends to infinite (positive of negative) when the variable tends to a value a. If the limit is +∞, it is written: lim f(x) = +∞. If the limit is -∞, it is written: lim f(x)= -∞ For example, in the graph f(x) = 1 (X-4)2
32. In the graph you can see that when x approaches to 4 by the left, the function tends to infinite, and in the same way occurs when approaching x to 4 by the right. This is, the function grows without stop. It is written: Lim 1 =∞ x4 (x-4)2
33. Write the limit with the symbol of infinite does not mean that the limit exists, since there is no real value for “L”, it only symbolizes that the function grows (or decreases) with unbounded behavior. Sometimes just some of the lateral limits tend to infinite, or both tend to infinite, but one to plus infinite and the other to less infinite; growths or decreases with no stop, without border.
34. For example, the graph of f(x) = 1 (x-4) In the graph you can see that when x approaches to 4 by the left, the function tends to -∞, and when x tends to 4 by the right, the function tends to +∞. In this case, you write: Lim 1 = -∞ x4 x-4 Lim 1 = +∞ x4 x-4
35. However, since the lateral limits are different you write lim1 = does not exist x4 x-4 to evaluate this class of limits, it is convenient to use the approximation method or to use the graph of the function. Observe that these classes of limits represent graphically a vertical asymptote.
36. 2.8.2 Limits in the infinite The limit in the infinite is the one where the variable tends to infinite (positive or negative) and the function tends to a value L. If x tends to +∞, it is written: lim f(x) = L. if x tends to -∞, it is written: lim = f(x) = L. x+∞ x-∞ The limit of algebraic function (polynomial) when x tends to infinite does not exists, since the function also tends to infinite positive or negative. For example, in the limit of the function f(x) = x+2, when x tends to infinite: Lim (x+2) x+∞ The solution is: lim (x+2) = (∞+2)= ∞ x+∞
37. The limit of a rational function can be zero; a value different from zero or well does not exists. In this class of limits, it is started with the fact that the limit of an expression where a constant is divided between a value that tends to the infinite, the result tends to zero. Lim k = 0 x∞ xn To evaluate this class of limits, where x tends to infinite, the method of approximate can be used, using the graph of the function, or well to transform the expression dividing it between the bases of greater exponent of the denominator in such a way that the previous limit can be applied.
38. EXAMPLE 1 Determine the limit of the function f(x) = (6x – 2)/(3x+3), when x tends to infinite. Lim 6x-2 x+∞ 3x+3 Solution by approximation: We have that when x tends to a +∞, the function tends to 2.
39. b) Solution using the graph In the graph you can see that when x tends to +∞, the function tends to 2.
40. c) Solution taking as reference Lim k =0 x∞ xn Each one of the terms of the expression is divided between the variable with greater exponent of the denominator (for this case it is divided between x): lim6x-2 = lim6x/x – 2/x x+∞ 3x+3 x+∞ 3x/x + 3/x It is simplified: lim6 – 2/x x+∞ 3 + 3/x
41. It is applied the limit of reference: lim6 – 2/x = 6-0 = 6 = 2 x+∞ 3 + 3/x 3+0 3 So lim6x – 2 = 2 x+∞ 3x + 3
