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It uses personal experience from studying Hardy (1908) and teaching calculus to argue that students should deliberately confront what seems visually impossible.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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origin?","Question",{"text":62,"@type":63},"The text explains that as x approaches 0, sin(1/x) begins to behave in an increasingly wild way, producing an oscillatory region that is hard to render accurately on paper.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"What three boundaries does the author identify?",{"text":67,"@type":63},"The author describes (1) a boundary between “nice” and “wild” functions, (2) a boundary between frequently studied versus more rarely examined functions, and (3) a boundary between drawable and undrawable behavior.",{"name":69,"@type":60,"acceptedAnswer":70},"How does the author suggest teaching sin(1/x) to students?",{"text":71,"@type":63},"Students should focus on graphing the composition of two known functions, analyze how 1/x maps inputs to outputs (including special points like x=1 and x=-1), and then connect that understanding to the behavior of sin(1/x).","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},598680,1792192584,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & 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surprising to students to learn that a natural combination of simple functions, the function sin(1/x), exhibits behaviour that is a great challenge to visualize. When x is large the function is relatively easy to draw; as x gets smaller the function begins to behave in an increasingly wild manner. The sin(1/x) function can serve as one of their first counterexamples, helping them to better appreciate the tamer functions that they normally encounter. I see three boundaries here. First, a boundary erected by mathematicians between‘nice’ versus ‘wild’functions-capturedfor example by the concept of continuity. Second, a boundary between those functions that are most often studied in calculus and pre-calculus classrooms, and those that are more rarely looked at. Third, the boundary between the drawable and the undrawable. In this example, we can witness this last boundary first-hand even as we attempt to sketch the curve. Yet, we can also continue the visualization in our mind’s eye beyond what we can represent on paper.  \nIntroduction  \nI t is possible to advance quite far in one’s mathematics  \neducation without encountering the function sin(1/x) . I describe it as an “encounter” because it felt that way tome when I read Hardy’s A Course of Pure Mathematics (1908) and reached Exercise 6 of Examples XV in Section 28: “Draw the graph of sin(1/x) .” I remember distinctly how strange it was at first to realize just how wild this function’s behaviour was near the origin. Unlike so many other graphs of functions of the sort that are treated either in upper-secondary school courses or beginningundergraduate courses, this function seemed to display anever-concentrating vigour and vitality inanever-narrowing neighbourhood. By stark contrast, the ‘nice’ functions I had spent most of my time learning about looked simpler  \nand simpler the more I zoomed in my visual field on any particular point on the graph. I thought I was familiar with sin(x) and with 1/x, but a simple combination of them held surprises for me, and I felt I was meeting with a stranger.  \nThe drawing I created on my page and the drawing on the following page in Hardy (1908) interested me greatly. These drawings captured some of the behaviour of the function but left out a region near x=0 that was impossible to render accurately. Hardy made the choice to leave the region blank; I continued drawing oscillations until I had nearly filled in with ink the region near x=0 and between-1 and 1. The latter choice is the one that Google makes, for what it is worth, when you search for sin(1/x) . The former choice is more honest, in the sense that it does not draw anything that is not accurate. In both cases, it seemed to me, the mind  \nneeded to continue an internal visual representation of an external visual representation that could not be continued. 1  \nI have taught introductions to calculus to various groups of students: Grade 12 students, first-year undergraduate students in the life sciences, first-year  \nundergraduate students in the physical sciences, inservice teachers who are taking a certificate program in mathematics. To all of these students I have introduced the graph of sin(1/x) . I believe that considering this curve and contemplating its properties can provide valuable insights to any student of mathematics. I explore some of these insights below; images of sin(1/x) appear just before the end.  \nI emphasize to students the following important feature of our task: we are graphing the composition of two functions that we are already well acquainted with. The function 1/x exhibits behaviour familiar from many situations, including, for example, pressure-volume dependence in ideal gases or price-quantity demanded in classical economics. I want students to realize that out of well-known, relatively familiar ingredients we can very quickly encounter an object tha","cbCaieRLAXFzUj35","https://ap.wps.com/l/cbCaieRLAXFzUj35","pdf",104094,"English","# Introduction\n## Encountering sin(1/x)\n## Three boundaries in visualization\n## Pedagogical framing and classroom activity","[{\"question\":\"Why is sin(1/x) difficult to visualize near the origin?\",\"answer\":\"The text explains that as x approaches 0, sin(1/x) begins to behave in an increasingly wild way, producing an oscillatory region that is hard to render accurately on paper.\"},{\"question\":\"What three boundaries does the author identify?\",\"answer\":\"The author describes (1) a boundary between “nice” and “wild” functions, (2) a boundary between frequently studied versus more rarely examined functions, and (3) a boundary between drawable and undrawable behavior.\"},{\"question\":\"How does the author suggest teaching sin(1/x) to students?\",\"answer\":\"Students should focus on graphing the composition of two known functions, analyze how 1/x maps inputs to outputs (including special points like x=1 and x=-1), and then connect that understanding to the behavior of sin(1/x).\"}]","Boundaries in Visualizing Mathematical Behaviour | PDF",1792087377,13]