12.1  Limits and Continuity of multi-variable functions

1-6. Find the limit, if it exists, or show that the limit does not exist.

1. .

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

(Sol)

0

0

Limit  exists and is 0.

 

 2.

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

(Sol)

   1

   -1

Limit does not exist.

 

3.  .

           http://matrix.skku.ac.kr/cal-lab/cal-12-1-3.html

(Sol)

0

0

Limit  exists and is 0.

 

4.  .

         http://matrix.skku.ac.kr/cal-lab/cal-12-1-4.html  

(Sol) 

0

0

Limit  exists and is 0.

 

5.  .

          http://matrix.skku.ac.kr/cal-lab/cal-12-1-5.html  

(Sol) 

0

0

Limit  exists and is 0.

 

6. .

          http://matrix.skku.ac.kr/cal-lab/cal-12-1-6.html  

(Sol) 

0

So


0

0

Limit  exists and is 0.


7-8. Let each of the following functions have the value 0 at the origin. Which of them are conti. at the origin?

 

7. .

              http://matrix.skku.ac.kr/cal-lab/cal-12-1-7.html   

(Sol)

0

0

Limit  exists and is 0.

So it is conti at the origin.

 8.

      http://matrix.skku.ac.kr/cal-lab/cal-12-1-8.html

(Sol)

 

   +Infinity

Limit does not exist.

So it is not conti at the origin.


9-10. Find and the set on which is continuous.

10. , .

(Sol)

  

 is continuous.

11. , .

(Sol)

    

   is continuous.