ANALYSIS OF APPROACHES TO THE STUDY OF LIMITS OF SEQUENCES AND FUNCTIONS AND THE USE OF INFORMATION TECHNOLOGIES

Summary. Article analyzes approaches to the study of limits of sequences and functions and the use of information technologies. The theorem on the limit of an intermediate sequence is of great importance when investigating the convergence of sequences and when calculating the limits of sequences. It is developed an algorithm for applying this theorem that is illustrated with examples. An example of calculating the limit of a function with a geometric interpretation of the obtained result is considered. The possibility of completing the calculation of limits at infinity, using the program code written by the author in the LaTeX programming language, is considered.


Introduction.
The themes of limits of sequences and functions are studied by students of various specialties in classical, technical, pedagogical, and technological universities in the course of mathematical analysis, in the section of mathematical analysis of the course of higher mathematics, in the course of applied mathematics.For students of different specialties, a different number of academic hours is given to study the themes of limits of sequences and functions.
It is especially often necessary to calculate the limits of sequences and functions for infinity.Infinities in scientific research are common [1]- [4].
It is convenient to prepare scientific results in the LaTeX system [5]- [20], which is particularly effective for creating complicated mathematical texts [21]- [22].Using the LaTeX system, one can perform calculations.
Main part and results.The theorem on the limit of an intermediate sequence (the theorem on three sequences) is of great importance when investigating the convergence of sequences and when calculating the limits of sequences.During a class with students, it is advisable to develop an algorithm for applying the theorem on the limit of an intermediate sequence (the theorem on three sequences): -Calculate the value of several members of a sequence.Hence, we can make assumptions about the convergence of the sequence and its limit.Example.Given the sequence with a common term Calculate the limit of this sequence as .Solution.The sum contains n terms, the smallest of which is the last, and the largest is the first.Hence we have the inequality as It is necessary to emphasize to students that it is convenient to use the equivalence ~ as where are real numbers, when calculating the limits of extreme sequences as .Since the limits of extreme sequences are equal to one, then the limit It is appropriate to emphasize to students that the limit of the given sequence can be calculated using the equivalence given above.Then we obtain Example.Given the sequence with a common term Calculate the limit of this sequence as .Solution.Evaluating the given sequence from below and above for , we obtain It is necessary to emphasize to the students that, when calculating the limits of extreme sequences as it is advisable to use the equivalence given above.Then the limit , n → SECTION XVIII.PHYSICS AND MATHEMATICS and the limit Since the limits of extreme sequences are equal to for and these limits are equal to one for , then, using the theorem on the limit of an intermediate sequence, we obtain It should be emphasized to the students that a known limit is used here.
It is necessary to draw the attention of students to the fact that the given limit can be calculated without using the theorem on the limit of an intermediate sequence, if one remembers the sum of the cubes of the first n natural numbers: Then, using the equivalence given above, we obtain It is advisable to emphasize to the students that it is possible to calculate the limit of the sequence with any predetermined precision.Using the limits of sequences, one can represent new unknown numbers or numbers that have no other representation (for example, one can represent irrational numbers in the form of limits of rational numbers).
It is necessary to note to students that if it is necessary to calculate the limit of a function and give a geometric interpretation of the result obtained, then the limit may be a finite number, may be equal to or , or may not exist.Students should remember the definition of the asymptotes of the graph of a function.
Let us illustrate this remark with the following example.
Example.Find the limit

Solution.
We have an uncertainty of the form , which we solve by getting rid of irrationality in the numerator, that is, by moving irrationality to the denominator and using the equivalence ~ as Thus, we obtain We interpret the obtained result geometrically: when , the graph of the function asymptotically goes to a straight line which is its horizontal asymptote.Conclusions.Thus, we have shown the peculiarities of calculating the limits of sequences and functions as .In cases where scientists in research need to calculate rather complicated limits as , it is preferable to calculate them analytically, as we have shown above.In similar cases, calculations on a computer as can be done only for large n.At the same time, the larger n is, the longer such calculations are performed, and there is no sense when calculating the limits on the computer as .However, in many cases, taking into account the possible difficulties when calculating the limits, it is possible to complete the calculation of limits as using computer programs.For example, given the sequence with a common term for , , .Then In this case, it is possible to complete the calculation of the limit on a computer using the programming code, for example, written by the author in the LaTeX programming language.In this programming code, the author created the new macro \sn: x → − 2 3 The given programming code generates the following result: Thus, the limit of the given sequence with the common term for , , , is equal to 52707600.