Limits of functions University of Sheffield（谢菲尔德大学功能的局限性）.pdfVIP

Limits of functions In this unit, we explain what it means for a function to tend to infinity, to minus infinity, or to a real limit, as x tends to infinity or to minus infinity. We also explain what it means for a function to tend to a real limit as x tends to a given real number. In each case, we give an example of a function that does not tend to a limit at all. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • decide whether a function tends to plus or minus infinity, or to a real limit, as x tends to infinity; • decide whether a function tends to plus or minus infinity, or to a real limit, as x tends to minus infinity; • decide whether a function tends to a real limit as x tends to a given real number. Contents 1. The limit of a function as x tends to infinity 2 2. The limit of a function as x tends to minus infinity 5 3. The limit of a function as x tends to a real number 8 c 1 mathcentre July 18, 2005 1. The limit of a function as x tends to infinity If we have a sequence (y )∞ , we can say what it means for the sequence to have a limit n n=1 as n tends to infinity. We write yn → l as n → ∞ if, however small a distance we choose, yn eventually gets closer to l than that distance, and stays closer. We can also write