VISUAL DETERMINATION OF FUNCTION LIMIT USING THE WALL METHOD

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 111 VISUAL DETERMINATION OF FUNCTION LIMIT USING THE WALL METHOD F.F. Turaev Associate Professor (Acting) at the Department of Mathematics and Physics, Alfraganus University (PhD) https://doi.org/10.5281/zenodo.18065444 Abstract. This study delves into the “wall method,” a visual and intuitive approach widely employed in mathematics to determine the limit of a function. The wall method demonstrates how a function’s graph becomes progressively “compressed” between two bounding lines as the input variable approaches a specific point. This visual squeezing effect provides a clear illustration of the value toward which the function converges, thereby making the abstract concept of limits more tangible and accessible. The method proves particularly valuable when analyzing functions that exhibit discontinuities or complex behavior, offering a geometric perspective that complements the formal epsilon-delta definition. By employing this approach, learners can better grasp the fundamental principles underlying limits, enhancing both conceptual understanding and visual intuition. Overall, the wall method serves as a powerful pedagogical tool, aiding educators in explaining intricate mathematical concepts and supporting students in developing a robust comprehension of limits. Keywords: wall method, visual approach, function limit, bounding lines, convergence, graph behavior, geometric interpretation, epsilon-delta concept, mathematical analysis, visualization method. INTRODUCTION. In mathematical analysis, the concept of a limit plays a fundamental role in understanding the behavior of functions near specific points. However, because limits are often introduced through abstract definitions, many learners find them difficult to visualize and comprehend intuitively. To bridge this gap, various graphical and conceptual tools have been developed, one of the most effective of which is the Wall Method. The Wall Method is a visual approach that illustrates how a function’s output becomes confined within an increasingly narrow region as the input approaches a particular value. By representing this process with two “walls” or bounding lines that move closer together, the method provides a clear geometric interpretation of convergence. It helps learners observe how a function “gets squeezed” toward a single value, making the idea of a limit more accessible. This method is especially beneficial in explaining discontinuous functions, sharp turns, or behaviors that are not immediately apparent through algebraic expressions alone. As such, the Wall Method serves as a powerful pedagogical tool for enhancing conceptual understanding and supporting students in connecting visual intuition with formal mathematical definitions. THE WALL METHOD The Wall Method is one of the visual approaches used in mathematics to determine the limit of a function. In this method, an imaginary wall is drawn on the graph at a specific point, and the curve of the function approaches this wall from both sides. By observing the function as it approaches from the left, one can see how the values change, and similarly, by observing from the right, the changes in values can also be tracked. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 112 This method makes it easier to determine the limit of a function at a point because it allows one to visually see how the curve approaches the point from both sides. 1Example Let us consider the following function H: 𝑓(𝑥)={2𝑥+2,𝑖𝑓𝑥 <1𝑡ℎ𝑒𝑛 2𝑥−4,𝑖𝑓𝑥 ≥1𝑡ℎ𝑒𝑛 Let us examine this example using the Wall Method: 𝑙𝑖𝑚 𝑥→1 𝐻(𝑥) does not exist but 𝑙𝑖𝑚 𝑥→−3𝐻(𝑥)=−4. 2Example Let us consider the function defined as follows: 𝐺(𝑥)={5,𝑖𝑓𝑥 =1𝑡ℎ𝑒𝑛 𝑥+1,𝑖𝑓𝑥 ≠1𝑡ℎ𝑒𝑛 a) Draw the graph of the given function and then find each of the specified limits (if they exist). In cases where the limit does not exist, provide a justification. b) 𝑙𝑖𝑚 𝑥→1𝐺(𝑥) b) 𝑙𝑖𝑚 𝑥→−2𝐺(𝑥) Solution The graph of the function G is shown below. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 113 a) If the values of x approach 1 from the left, the values of 𝐺(𝑥) approach 2. Therefore, the left-hand limit is 2. Similarly, when x approaches 1 from the right, the values of 𝐺(𝑥) also approach 2. Hence, the right-hand limit is 2 as well. Since the left-hand and right-hand limits are equal—both being 2—we have: 𝑙𝑖𝑚 𝑥→1 𝐺(𝑥)=2 Note that the limit is equal to 2, but this is not the same as the actual value of the function at x = 1, since 𝐺(1)=5 Numerical approach Graphical approach b) Based on the method applied in part (a) above, we obtain the following result. 𝑙𝑖𝑚 𝑥→−2𝐺(𝑥)=−1 In this example, the limit is equal to –1, and this value corresponds to the actual value of the function at x = –2: G(–2) = –1. Numerical approach Graphical approach SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 114 CONCLUSION The numerical method and the graphical method are effective approaches for determining the limit of a function. Using the numerical method, one can observe how the function values change by approaching the limit point with successive values of the argument. The graphical method, on the other hand, visually illustrates how the function approaches the limit point by plotting its graph. Additionally, the Wall Method allows for a more intuitive understanding of limits, as it shows how the curve approaches the point from both sides, making it easier to determine the limit value. As seen in the examples, the existence or non-existence of a limit can be clearly identified using both numerical and graphical approaches. Overall, these methods greatly assist students in gaining a deeper understanding of the concept of limits and in developing analytical thinking and problem-solving skills in mathematics. REFERENCES 1. Canuto, C., & Tabacco, A. (2015). Mathematical Analysis 1. Milan, Italy. 2. Baumann, G. (2010). Mathematics for Engineers I. Munich, Germany. 3. Khurramov, Sh.R. (2018). Higher Mathematics, Vol. 1–2. Tashkent: Tafakkur Publishing. 4. Soatov, Y.O. (1996). Higher Mathematics, Textbook, Vol. 1–3. Tashkent: Uzbekistan Publishing. – 640 p. 5. Tojiev, Sh.I. (2002). Solving Problems in Higher Mathematics, Textbook. Tashkent: Uzbekistan Publishing. – 512 p.