DETERMINING THE LIMIT OF A FUNCTION USING NUMERICAL AND GRAPHICAL METHODS

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 115 DETERMINING THE LIMIT OF A FUNCTION USING NUMERICAL AND GRAPHICAL METHODS F.F. Turaev Associate Professor (Acting) at the Department of Mathematics and Physics, Alfraganus University (Ph.D) https://doi.org/10.5281/zenodo.18065477 Abstract. This work discusses the numerical (table) method and the graphical method used to determine the limit of a function, along with their procedures and advantages. In the numerical method, the values of the function are analyzed as the argument approaches the limit point, allowing the behavior of the function and its convergence toward the limit to be observed. In the graphical method, the graph of the function is constructed, and the behavior of the graph near the limit point is examined to determine whether the limit exists and to identify its value. A comparative analysis of both methods highlights their practical significance and their role in enhancing intuitive understanding of the concept of limits. This topic contributes to improving visualization in explaining function limits and helps develop analytical and independent thinking skills in students. Keywords: function limit, numerical method, table method, graphical method, limit calculation, function graph, convergence, boundary value, mathematical analysis, visual method, limit point, function behavior. INTRODUCTION. One of the most important concepts in mathematical analysis is the limit of a function. A limit studies the value that a function approaches as its argument nears a particular point. This concept plays a key role not only in mathematical theory but also in many applied fields such as physics, engineering, economics, and computer science, where it is essential for analyzing continuity, rates of change, and boundary behaviors. There are various methods for determining the limit of a function, among which the numerical (table) method and the graphical method are the most illustrative and comprehensible approaches. The numerical method allows one to observe how the function changes by examining successive values of the argument as it approaches the limit point. The graphical method, on the other hand, provides a visual analysis of the function’s behavior through its graph. This topic explores the principles, application procedures, advantages, and limitations of these two methods. It also helps in developing a deeper understanding of the concept of limits, enabling students to analyze function behavior, enhance logical thinking, and strengthen independent research skills. One of the important aspects of analysis is examining how the values of functions change with respect to changes in their arguments. The foundation of this study is the concept of a limit. Suppose we have a function 𝑓(𝑥). If the values of xxx approach a certain number aaa, and at the same time the values of 𝑓(𝑥) approach another number 𝐿, then we say: “the limit of 𝑓(𝑥)as x approaches 𝑎 is 𝐿”. This means that no matter how close the values of 𝑥 get to a, the values of 𝑓(𝑥) get correspondingly close to 𝐿. For example. Let us calculate the limit of (𝑓(𝑥)= 2𝑥 + 3). When the values of (𝑥) approach 4, that is, as (𝑥) approaches 4 from both the left and the right, the values of the function SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 116 (𝑓(𝑥)) also approach 11. Therefore, as (𝑥) approaches 4 (from either side), the value of (𝑓(𝑥)) gets close to 11. Numerical approach Graphical approach 𝑥 → 4 means that the number x is approaching 4 from both sides. In this case, 2𝑥 + 3 → 11 means that as the value of x gets closer to 4, the expression 2𝑥 + 3 approaches 11. The number 11 is the exact and unique limit that the expression 2x + 3 approaches as the value of x gets closer to 4 from both sides. That is, as x approaches 4, the value of the expression gradually approaches 11 and ultimately becomes exactly equal to 11. This can be expressed using mathematical notation as follows:𝑙𝑖𝑚 𝑥→4(2𝑥 + 3)=11 It is read as: The limit of 2𝑥 + 3 as x approaches 4 is 11. Definition 1. If the values of x approach a point a (but never exactly equal a), and the values of the function f(x) approach a certain number L, then: 𝑙𝑖𝑚 𝑥→𝑎 𝑓(𝑥)= 𝐿, It is written as and interpreted as: ‘The limit of f(x) as x approaches a is L. Here, L must be a unique and real number. If we want to specify the direction from which we are approaching, we use a special notation in mathematics. If x is approaching the point a from the left (that is, through values less than a, x < a), we write: 𝑙𝑖𝑚 𝑥→𝑎−𝑓(𝑥) or If x is approaching the point a from the right (that is, through values greater than a, x > a), we write: 𝑙𝑖𝑚 𝑥→𝑎+𝑓(𝑥) The input values are approaching 4 from the left The input values are approaching 4 from the right The values of the function are approaching 11 The function values are approaching 11 SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 117 These limits are called the left-hand limit and the right-hand limit, respectively. Let us once again consider the function f(x) = 2x + 3. We have shown that as x approaches 4, the values of the function approach 11. This can be expressed as follows: 𝑙𝑖𝑚 𝑥→4(2𝑥 + 3)=11 You might be wondering, Why don’t we just substitute 4 into the function to get 11? This reasoning is partially correct (and in some cases, we will see that such shortcuts can be used), but it is important to remember that here we are not interested in the exact value at x = 4. Instead, we are concerned with how the function values change when x is very close to 4—that is, the behavior of the function 𝑓(𝑥)= 2𝑥 + 3 as x approaches 4. Now, it is useful to briefly summarize what we have learned so far.  At x = 4, the value of the function is 11, and this is represented on the graph of f as the point (4, 11).  For values of x close to 4, the values of f(x) also approach 11 accordingly—this is exactly the limit we are examining. The concept of a limit allows for a clearer understanding of certain properties of functions. Let us consider the following example. Example 1. 𝑓(𝑥)=𝑥2−1 𝑥−1 a) What is the value of f(1)? b) Find the limit of the function f(x) as x approaches 1. Solution. a) The solution does not exist because the denominator becomes zero when calculating the fraction, which is mathematically undefined. 𝑓(1)=12−1 1−1 =0 0 Thus, 𝑓(1) does not belong to the domain of the function. Therefore, the function does not take any value at this point, resulting in a discontinuity around 𝑥 = 1. By carefully analyzing the graph provided below, we can see that at 𝑥 = 1 the function's curve is broken or not marked with any symbol, which clearly indicates that the function is undefined at this point. From the graph, it is evident that as x approaches 1 from the left or the right, the function’s value may exist; however, at that specific point, there is no definite value. For this reason, 𝑓(1) cannot be calculated, and we say it does not exist. This situation indicates a discontinuity in the function, often referred to as a removable discontinuity or a “hole.” SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 118 Numerical approach Graphical approach b) We select values of (x) close to 1 from both sides (left and right) — see the table above — and observe that the function values gradually approach 2. Therefore, the limit of the function (f(x)) as (x) approaches 1 is as follows: 𝑙𝑖𝑚 𝑥→1 𝑓(𝑥)= 2. On the graph, there is a “discontinuity” at the point (1,2), meaning that the function is undefined at this point. Nevertheless, since the function values approach the single definite number 2 as 𝑥 → 1, we can say that the limit exists. Note: From a mathematical point of view, the existence of a function’s limit at a certain point does not depend on the value of the function at that point. Even if 𝑓(𝑎) does not exist at all, the limit may still exist, and vice versa. Therefore, whether a limit exists at 𝑥 = 𝑎 is independent of whether 𝑓(𝑎) is defined or not. This illustrates that the concepts of a limit and the function’s value are independent of each other. CONCLUSION Calculating the limit of a function using numerical and graphical methods is one of the important aspects of mathematical analysis. The numerical method allows for the analysis of how function values approach the limit point using precise numbers, while the graphical method visually represents this process, helping to understand the overall behavior of the function. Both methods complement each other: the numerical method provides accuracy and consistency, whereas the graphical method enhances intuitive understanding. Thus, these approaches are effective tools for fully comprehending the limiting behavior of functions, developing mathematical thinking, and solving practical problems. REFERENCES 1. 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