# The Abstraction Process of Limit Knowledge

## Article excerpt

Limits are a basic and important concept in mathematics. They can be assumed as one of the most fundamental concepts and influential instruments of general mathematics because they lead especially to the understanding of derivations and integrals (Çildir, 2012; Çoker, Özer, & Taç, 1989). In other words, limit knowledge is a pre-concept related to many important mathematical concepts such as derivatives, integrals, continuities, and convergences. Therefore, it also constitutes an important role in high school and college mathematics curricula.

Knowledge and Historical Development of Limits

Limits were used in theorems related to shapes with curvilinear sides in early times. Euclid and Archimedes used this concept for the first time. For example, Archimedes used them to calculate the area of a circle. In his justification, the area of the circle is equal to the area of a right triangle whose base equals the circle's circumference and whose height equals the circle's radius. In this process, the circle is assumed to be a straight line equal to the circumference of it. He calculated the inferior and superior limits to find the straight line in this justification (Cajori, 2014, p. 25). However, the use of limits with the same current meaning was based on work from the 17th century. For example, Fermat examined the interrelation between the limit of a curve at a point on the graph and the tangent of that point of the curve (Baki, 2008, p. 147). Afterwards, the researchers Newton and Leibniz used limits to calculate integrals. In other words, differential equations emerged from knowledge on limits (Baki, 2014, p. 145). Newton can be said to have used this concept in general, not for any specific purpose. He used it to extrapolate a few different principles used for a variety of purposes (Cajori, 2014, p. 234). Similarly, the area under the curve of y = x2 restricted for 0 < x < 1 was shown to be 1/3 by Cavalieri using limit knowledge (Baki, 2008, p. 146). Meanwhile, the definition of the limit was not given until 1817 by Bolzano. Additionally, Weierstrass and Cauchy indicated limits were applied in calculus nineteenth century (Arslan & Çelik, 2015, p. 483; Baki, 2008, p. 149). Formal and informal definitions can be given for limit knowledge. In calculus, the epsilon-delta definition of limit is a formalization of the notion of limit (Balci, 2014).

Difficulties in Learning and Teaching Limits

These days, difficulties and misconceptions about limit knowledge are encountered by students as learning the subject of limits brings along serious difficulties for students in the advanced subjects of high-school curriculum. Because limits contain different procedures that include infinity in particular, they are also not a simple subject (Baçtürk & Dönmez, 2011; Tangül, Barak, & Ôzdaç, 2015). Most students cannot place the knowledge of limits in their minds and for this reason have difficulty making sense of it; as a result, students have trouble learning limit knowledge (Çildir, 2012). Specifically, they have difficulty performing operations related to limits. This situation has been reflected in different studies. Within this context, Durmuç (2004); Gürbüz, Toprak, Yapici, and Dogan (2011); and Tatar, Okur, and Tuna (2008) aimed to determine difficult subjects in mathematics courses and to reveal the roots underlying these difficulties. Thus all of these studies found the subject of limits taking place at the top of subjects which students perceive as difficult. The desire to uncover the difficulties with limits is the main reason why the number of studies made on this concept has increased (Baçtürk & Dönmez, 2011).

Some limit-knowledge studies (Barak, 2007; Baçtürk & Dönmez 2011; Denbel, 2014; Juter, 2006; Szydlik, 2000; Tangül et al., 2015) revealed students' difficulties and misconceptions. In the studies by Barak (2007) and Baçtürk and Dönmez (2011), students were determined to have different misconceptions about limit knowledge. …

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