Calculus-Sec-2-1-Solution


  

    2.1    Limits of Functions                 by SGLee - HSKim- SWSun

                                                                                                           http://youtu.be/LZSmRPAAXME 

 

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

1. 

  

    




2. 

   

    If exists, then , when  approaches to 0

  (1)

       

  (2)

     , so  than  does not exist when  approaches to 0.   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.  

 

    Since  when 




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.




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

          

   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 .

    Take, 

 

   (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)   ;  

       ⅱ)  ( is positive integer) ;  

       ⅲ)  (  is positive integer) ;  

       ⅳ)   ;  

       Therefore 

       When .




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

  http://matrix.skku.ac.kr/cal-lab/cal-2-1-15.html

     , 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.  

           [Find ] Let . If , then = = .

        [Side calculation]   . ■




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. 

      [Find ] Let   = . If || < , then  .

  [Side calculation] .    ■




24. 

 

      [Find ] Let . If , then =

       == = .

      [Side calculation]  . ■




25. 

 

          [Find ] Let {}. If , then = 

        [Side calculation]  . ■




26. 

 

      [Find ] Let . If , then =

     

     [Side calculation]    Take . If , then =.  ■




27. .

http://matrix.skku.ac.kr/cal-lab/cal-2-1-27.html

 

      [Find ]  Let  . If , then

    =    . 

     [Side calculation]   and   . ■




28. 

http://matrix.skku.ac.kr/cal-lab/cal-2-1-28.html

 

    , [Find ] Let   (). If , then  .

    [Side calculation] Since    . ■




29. 

 

    Given any (large) number  we must 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 .




30. 

 

        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.




31. 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 




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

   (a) 

   (b)  if 

   (a) 

   (b) 




                                                                            

                

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