[list][*]Subject: Mathematical analysis[/*][*]Class: XI[br][/*][*]Duration: 50 minutes[br][/*][*]ICT tools: teacher's computer with projector, student computer, tablets or smartphones.[i][br][/i][/*][/list]

Calculate the limits of polynomial and rational functions, in the indeterminate cases [math]\infty-\infty[/math], [math]\frac{\pm\infty}{\pm\infty}[/math].

[justify]At the end of the lesson, students will be able to:[/justify][list][*]calculate the limits at + and - infinity, of the polynomial functions in the indeterminate case [math]\infty-\infty[/math];[/*][*]calculate the limits at + and - infinity of the rational functions in the indeterminate case [math]\frac{\pm\infty}{\pm\infty}[/math].[/*][/list]

[justify]During the lesson, students will become able to:[/justify][list][*]to identify the indeterminate cases when calculating polynomial and rational functions limits;[/*][*]to apply forcing the common factor method to calculating polynomial limits;[/*][*]to identify the limit at infinity, of a polynomial function, with the limit of the maximum degree term;[/*][*]to apply forcing the common factor and simplification method to calculating limits at infinity of the rational functions;[/*][*]to identify the limit at infinity of a rational function with limit of the maximum degree terms ratio.[/*][/list]

[i]Strategies and methods: [br][br][/i]Limits of polynomial and rational functions are among the first studied within the Limits chapter, and students are not yet familiar with their calculation, especially in exempt cases. Being in their first attempts to calculate the limits of the functions, especially in the excepted cases, we will provide them with methods and means for reducing the abstract character of Calculus and improve understanding and retention.[br][br]In order for this strategy to be effective and to result in the understanding and setting of concepts for all students, it is necessary, to see and study many graphs of functions, that would be impossible to draw on the blackboard, in the time of one class hour. However, the students do only know how to draw the graphics of elementary functions, and, barely towards the end of the course, will be able to represent composite functions. However, they trust computer technology, and if we anticipate presenting these graphics, pupils not only see in them a purpose with real finality, but they can observe and understand the behavior of functions at infinity through a synthetic view, which creates the feeling of being known and concrete. [br][br]The methods used include GeoGebra modeling functions and demonstration of standard calculation methods (forcing the common factor and simplification). The interactive applications will be used throughout the entire lesson, alternating with individual, written solutions.[br][br][i]Lesson activities:[/i][br][list][*]The theme and objectives of the lesson are announced and discussed; some exempted cases are recapitulated;[/*][*]The problem sheets are shared, digital devices open, GeoGebra interactive applications, "Polynomial limits at infinity" and "Rational limits at infinity" open;[/*][*]The main task of the students is to solve a share of the exercises on the printed sheets (the rest remaining as homework); [/*][*]Frontally, it will be explored and proven on the blackboard, that the limit at infinity of the polynomial function is equal with the limit of the maximum degree term, eliminating the case of indetermination, by forcing a common factor;[/*][*]Similarly, we provide the proof that the limit at infinity of the rational function is equal to the limit of the ratio of maximum degree terms, eliminating the indeterminate case, by forcing a common factor, and simplifying the fraction;[/*][*]We invite students to use the provided applications, to visualize the polynomial / rational functions studied, but also to compare their limits at infinity with those of the maximum degree terms, or their ratio;[/*][*]Students will calculate the limits and will immediately check the results with the GeoGebra applications.[/*][*]Applications can also be used at home, even on phones, for further learning and solving the homework.[/*][/list]

[justify]Interactive worksheet[br]Limits at infinity of the polynomial functions: [url=https://ggbm.at/Q5FS8vwv]https://ggbm.at/Q5FS8vwv[/url][br][br]Interactive worksheet[br]Limits at infinity of the rational functions: [url=https://ggbm.at/q8xBnyPA]https://ggbm.at/q8xBnyPA[/url][br][br]Printable worksheet (pdf)[br]Limits at infinity of the polynomial and rational functions:[br]https[url=https://www.geogebra.org/worksheet/edit/id/zNEjgD8W]://www.geogebra.org/worksheet/edit/id/zNEjgD8W[/url][/justify]

Digital devices (minimum phones) with GeoGebra Graph Calculator will be used throughout the clock, depending on student choice. However, the lesson can be maintained without them, provided the drawing board and the power function boards (if any) are used.