Katedra matematiky - Katolícka univerzita v Ružomberku

152 Zbigniew Powązkadifferent notation methods of neighborhoods of unlimited elements whichbelong to extended set of real numbers.The next step contains exercises of finding neighborhoods of any pointof numerical axis, which are part of inverse image of appropriate neighborhoodsof a functionf. After these didactically operations one defines generalneighborhood-based definition of limit of a function at a point. Thereafterthis general definition is specified according to particular circumstances.In this development definitions of finite and infinite limit of the sequence,correspondingly in R and R ∪ {−∞,+∞} are particular cases of generaldefinition of function limit.Example 3.Interesting didactically solution for introducing the concept of limit ofthe function at a point is to prior define the concept of the function continuity.Then, investigation if the function has a limit in point x 0 amountsto give answer to the question about existence of continual protraction ofgiven function to the set which is domain of this function extended by pointx 0 . This idea is described in textbook by I. Kluvánek (2007). ). Examplesof tasks which could be use for functional develop of this concept are containedin book by J. Gunčaga, J. Fulier, P. Eisenmann (2008). We can alsofind there results of researches upon didactically mileage of this method.Example which appears below presents different idea of introducing theconcept of limit of the function at a point. It is less time-consuming but itrequires use of some concepts which do not appear in current curriculum ofmath teaching in middle school.Example 4.Different conception of introducing definitions of limit and continuity ofthe function is earlier development of metric in unlimited set concept andafterward development of concept of convergence to the metric.In this case, at the level of concrete activities there should be worked outtask in which there are examined examples of different metrics, in particularmetrics in sets R and R∪{−∞,+∞}.Then, there should be defined convergence to metric, in general and inparticular cases. These deliberations can be conceded as concrete - imaginaryactivities.At level of abstract activities there are achieved proofs of theorems relevantto properties of the concept of function limit which are true in allmodels. Then, there are discussed particular cases.Each of these solutions has advantages and disadvantages. Choice of theparticular didactically situation should depend on meritorical backgroundof students and pupils.