Infinite Limit: Definition, Questions and Discussion, and History

Infinite Limit – Does Sinaumed’s like math? If so, what material is your favorite, is it algebra or limits? Algebra and limits are different , yes , even though both of them have variables and various numbers in them. But don’t think that the material in mathematics is just calculations, but can also be applied in everyday life, you know… Then, what is an infinite limit? Who are the great figures who find limits in everyday life? What is the concept and nature of a limit? Come on, see the following reviews!

The Concept and Nature of the Infinite Limit

A simple concept so that Sinaumed’s understands what a limit is, you can take the following example. When at a shop, try to take the candy in the jar by holding it. On the first grip, you get 5 candy wrappers. Then on the second grip, get 6 candy wrappers. At hand when, get 5 candy wrappers. Then on the fourth grip, get 7 candy wrappers. Finally alias on the fifth grip, get 6 candy wrappers. So, the average of the five grips is 5 packs and is almost close to 6. Now, “almost close” is what is analogous to the concept of limit.

Infinite Limit is a limit concept involving the symbols ∞ and -∞, that is, if the value of the function f(x) increases or decreases without limit or if x increases or decreases without limit. The first concept is about the limit of the function f at point c for the function f which is limited to an interval containing c. While the second concept is about the limit of the function f for a variable x that increases without limit (x→∞) or for a variable x that decreases without limit x→-∞), which is known as an infinite limit. 

Then, the properties of the limit at one point and the limit of the composition function for functions that have limits, and the principle of clamping also applies to limits at infinity. The statement of the theorem is exactly the same, but x→c is replaced by x→∞, or replaced by x→ – ∞, and the origin of f is adjusted.

In short, this infinite limit is a form of study to find out the trend of a function, if the value of the variable is indeed made bigger. If it is said that x goes to infinity, then it will be written as (x→∞). That is, the value of x will get bigger or bigger until it is unlimited.

Infinite Limit Formula

To calculate the tendency of a function that is made bigger, of course you have to use a certain formula. Reporting from edmodo.id , an infinite limit has its own formula depending on its shape.

Infinite Limit Formula with Polynomial Form

The polynomial form in variable X to the highest power of one, if depicted in a Cartesian diagram, will form a straight line. Well, the limit value in the polynomial form really depends on the highest power of the polynomial. The limit of a function that has a variable x, will have a direct effect on the function f(x).

Infinite Limit Formula in Fractional Form

Infinity Limit Formula in Trigonometry Form

Infinite Limit Questions and Discussion

Example Question 1

DISCUSSION

Example Problem 2

DISCUSSION

Example Problem 3

DISCUSSION

Know the History of Limits

Before discussing what an infinite limit is, it would be nice if Sinaumed’s knew the history of limits first. Basically, the limit of a function is a misconception in calculus and analysis regarding the behavior of a function as it approaches a certain input point. A function will later map the output f(x) for each input x. The function has a limit L at the input point p, when f(x) is “approaching” to L, as well as when x is close to p. In other words, f(x) gets closer to L, as x also gets closer to p.

If analyzed further, if f is applied to any input that is close enough to p, then the result is an output that is (arbitrarily) close to L. Well, if the input that is close to p turns out to be mapped to very different outputs, then the function f It can be said that there is no limit.

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Has Sinaumed’s ever wondered what the history of the existence of limits is? Well, it turns out that this definition of limit has been studied since the 19th century, you know…

It should be noted that the history of the development of calculus can be seen from the period of ancient times, medieval times, and modern times. In the ancient period, some ideas about integral calculus arose, but unfortunately it was not developed properly and systematically. The reason was simple, because at that time there was a lack of knowledge or references related to calculus. Calculations of volume and area which are the main functions of integral calculus have been traced to and preserved in an Egyptian Moscow Papyrus (1800 BC). Here’s a little trivia, papyrus is a manuscript of material that resembles thick paper and is usually used for writing media in ancient times. Well, the Moscow Papyrus stated that the Egyptians had been able to calculate the volume of the pyramid which was later developed again by Archimedes and at the same time created a heuristic that resembled integral calculus.

Continuing in the Middle Ages, an Indian mathematician named Aryabhata used the concept of infinity in 499 and at the same time expressed matters relating to astronomy in the form of basic differential equations. This equation was developed again by Bhaskara II in the 12th century to become the initial form of the derivative which represented infinitely small changes. This is also the initial form of Rolle’s Theorem.

Around 1000, there was an Iraqi mathematician named Ibn al-Haytham, who became the first person to derive a formula for calculating the sum of the powers of four using mathematical induction. Ibn al-Haytham also developed a method for deriving the general formula from the product of the power of the integral, which of course became important in the development of integral calculus. Continuing in the 12th century, a Persian mathematician named Sharaf al-Din al-Tusi emerged who managed to find the derivative of cubic functions which became important in differential calculus.

Meanwhile, in modern times, independent discoveries emerged, precisely at the beginning of the 17th century in Japan, which were initiated by a mathematician named Seki Kowa. Different countries, different mathematicians who sparked their discovery of limits. In Europe, there are several mathematicians who have made breakthroughs in calculus material, for example, there are John Wallis and Isaac Barrow. Even James Gregory also helped prove calculus with a special case of the fundamental theorem of calculus in 1886 to be exact.

Some of the other notable experts who encouraged the discovery of this calculus are Leibniz and Newton. These two experts are considered to be the inventors of calculus separately but at about the same time. Newton applied calculus in general to physics, while Leibniz developed calculus in everyday life. So, when Leibniz and Newton succeeded in publishing the results of their research for the first time, a controversy arose which “debated” about which mathematician was more deserving of the award. Newton is considered to have completed his research results first, but Leibniz was the first to publish them. In fact, Newton accused Leibniz of stealing his ideas through his notes, which at that time he often lent to several members of the Royal Society.

So, to solve this problem, a detailed examination was carried out to show that the two mathematicians were indeed working separately, with Leibniz starting from integrals, while Newton started from derivatives. After the examination, both Newton and Leibniz were declared mathematicians who played a major role in the field of calculus and received awards. Leibniz is considered to be the person who gave the name to this branch of mathematics namely “Calculus”, while Newton is considered to be the figure who gave the name “The Science of Fluxions”.

Since then, many mathematicians have contributed to the development of calculus, one of which is Maria Gaetana Agnesi in 1748. Maria discovered further developments from calculus in the form of finite and infinitesimal analysis. Apart from that, there is also Cauchy who also discusses limits in his research entitled Cours d’analyse in 1821 and is considered a standard method for explaining limits.

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In general, this calculus was developed by manipulating very small quantities, of course the objects are numbers. An infinitely small dx number can be greater than 0, but can also be smaller than any number in the sequence 1, ½, ⅓, and so on; and any positive real number. Also, any multiplication by an ‘infinitely small’ number is still ‘infinitely small’. In other words, this infinitesimal does not satisfy Archimedes’ point of view. Therefore, calculus is a set of techniques for manipulating ‘small infinity’.

In the 19th century, the concept of ‘infinitely small’ was abandoned as unconvincing and replaced by the concept of limit. Limit describes the value of a function at a certain input value with the result of the closest input value. It is from this point of view that calculus is a set of techniques for manipulating certain limits.

If we analyze it again, the definition of the limit of a function is: “Given a function f(x) which is defined at intervals around p, with the possible exception of p itself. We say that the limit f(x) when x approaches p is L ”, then the writing is:

Scientists Contributing to Limits

  1. Augustin-Louis Cauchy 

Previously, Sinaumed’s must have realized that Cauchy’s name appeared in the history of Limit’s emergence. Yep, he was born on August 21, 1789 in Paris, France, and then died on May 23, 1857 at the age of 67, which was known as a French mathematician. Cauchy pursued his education at the Ecole Polytechnique, because his health was not so good, he made a career as a professor at the Ecole Polytechnique and the College de France.

Although calculus was invented around the 17th century, its basics remained muddled and messy until Cauchy and his colleagues conducted further research.

  1. Sir Isaac Newton

Newton, besides being a physicist, was also an English mathematician, astronomer, natural philosopher, to theologian, who was born on January 4, 1643 and died on March 31, 1727 at the age of 84. He was a follower of the heliocentric school and became the most influential scientist in history. Even Newton is also called the “Father of Classical Physics”.

His book Philosophiæ Naturalis Principia Mathematica , published in 1687, is considered the most influential book in the history of science, because it discusses the foundations of classical mechanics. In the book, Newton helped to describe the law of gravity and the three laws of motion that dominated the scientific view of the universe for three centuries. Newton also managed to show that the motion of objects on Earth and other celestial bodies is governed by the same set of natural laws. Newton became a figure who was able to prove the relationship between Kepler’s laws of planetary motion and his theory of gravity.

  1. Gottfried Wilhelm Leibniz

Leibniz is a German philosopher born on July 1, 1646 who is famous for the Theodicee concept he created. This understanding reveals that humans live in the best possible world because this world was created by a perfect God. Apart from being a philosopher, he is also a mathematician, diplomat, physicist, historian, and a doctorate in church law. His contribution to science was enormous, so many journals and manuscripts were published under his name.

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