2.1 Limits of functions

1-4. Find the following limits or explain why the limit does not exist.

1.

 

   


2.

     does not exist


3.

http://matrix.skku.ac.kr/cal-lab/cal-2-1-3.html 

   We draw with the following Sage command. We see it diverge    (  ) as  .

  


  


   We can find with the following Sage command.

  

   +Infinity


4.  

 



5.

   http://matrix.skku.ac.kr/cal-lab/cal-2-1-5.html 

 


6.  

   http://matrix.skku.ac.kr/cal-lab/cal-2-1-6.html




 7.

   http://matrix.skku.ac.kr/cal-lab/cal-2-1-7.html

   We draw with the following Sage command. We see the function converge to as  .   

   

   

   We can find with the following Sage command.


   


0


8. The sign function, denoted by , is defined by the following formulas


          


   Find the following limits or explain why the limit does not exist.


 (a)

      (b) does not exist ( )

      (c)

      (d)


9. Consider the function .

   (a) Find and .

 


   (b) Find the asymptotes of ; vertical, horizontal, vertical and oblique.  (For the oblique asymptote, find the straight line which is closer and closer to as )


 : vertical asymptote

       : oblique asymptote


   (c) Sketch the graph.


10. Draw the graph of a function with all of the following properties:

   (a) its domain is

   (b) there is a vertical asymptote at

   (c)

   (d)

   (e) does not exist.

   (f) does not exist.

   (g)


11. Let .

   (a) Find or explain why the limit does not exist.

 

   (b) Find and such that for all .

 Since , we have .

        so,

   (c) Use the Squeeze Theorem to find .

 We know that and , so by the Squeeze Theorem,


12. Use the Squeeze Theorem to find     .

   http://matrix.skku.ac.kr/cal-lab/cal-2-1-12.html  

   


   



13. Use squeeze Theorem to find

   http://matrix.skku.ac.kr/cal-lab/cal-2-1-13.html 

   

   


   (※ is not always smaller than , but it is when near 0. So the Squeeze Theorem can be used.)

14. Let . Find all positive integer such that

 


       ⅰ) n=4  ; 

       ⅱ) n<4  ; 

       ⅲ) n>4  ; 


       Therefore


15. Find all the asymptotes (vertical, horizontal, and oblique) of the function .

 , so ,, and Thus, x=-1 and x=2 are vertical asymptotes.


       , Thus, is an oblique asymptote.


16. Find the limit

 


17. Consider .

   (a) Find all the vertical asymptotes for .

 , so .

      Thus, are vertical asymptotes.

   (b) If we restrict the domain to , then show that there exists an inverse function defined on .

   (c) If the above inverse function is , then find all the horizontal asymptotes.


18. Find such that whenever .

19. Use an argument to prove that .

 Let be a given positive number. Here and . Claim is to find a number such that      whenever . With easy computation, we may choose to get the desired result.


20. Use the argument to prove that if .

 Let be a given positive number. Here and . Claim is to find a number such that whenever . With an easy computation, we may choose to get the desired result.


21-26. Prove the statements using an argument.


21.       

 Let be a given positive number. Here and . Claim is to find a number such that whenever . With an easy computation, we may choose to get the desired result.


22.

 Let be a given positive number. Here and . Claim is to find a number such that whenever . With an easy computation, we may choose to get the desired result.


23.

 Let be a given positive number. Here and . Claim is to find a number such that whenever . With an easy computation, we may choose to get the desired result.


24.

 Let be a given positive number. Here and . Claim is to find a number such that whenever . With an easy computation, we may choose to get the desired result.


25.

 Given any (large) number to find such that      whenever      . Since both     and    are positive,   whenever .

             Taking the square root of both sides and recalling that , we get whenever .

       So for any , choose .

       Now if , then , that is,.

       Thus whenever .

       therefore .


26.

 Let be a given positive number. Here and . Claim is to find a number such that whenever . With an easy computation, we may choose  to get the desired result.


27. Use an argument to prove that

 Given any (large) number to find such that whenever .

       Since both and are positive, whenever .

       So for any , choose .

       Now if , then

       Thus


28. If and , where is a real number. Show that

   (a)

   (b) if

   (a)

   (b)