13.2

13.2    Limits and Continuity of Multivariable Functions

by SGLee - HSKim, VLang, KHKim 1-7. Find the limit, if it exists, or show that the limit does not exist.

1.  <Solution from Cal-Book>

Let and along the line . So the limit of  does not exist. For example, for  and for  .

<Detailed Solution>

Let , so that when .  Therefore the limit,

.

The limit does not depend on  or ; thus, there is no limit.

2. .

<Solution from Cal-Book> As and   .

The limit exists, and it is .

<Detailed Solution>

Let  and , so that  when . Therefore the limit,

The limit does exist, and it is 0.

3. http://matrix.skku.ac.kr/cal-lab/cal-12-1-Exs3.html

<Cal-Book Solution>

Answer: The limit does not exist.

<Detailed Solution>

Let , so that when .  Therefore the limit,

The limit does not depend on  or , thus, there is no limit.

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

<Cal-Book Solution>

After plotting the function, we know it converges. So the order of and does not matter.

<Detailed Solution>

5.  Let and along the line .

The limit does not exist.

6.  After plotting the function, we know it converges.

Let and along the line .

7.  So .

The limit exists, and it is .

8. What value of for will make the function continuous at .  => Since , then by Squeeze Theorem.

Hence, if we define  will be continuous at .

9-10. Let each of the following functions have the value 0 at the origin.

9.  The limit exists and it is .

So is continuous at the origin. 10.  The limit does not exist.

11-12. Find and the set on which is continuous.

11.   Since is continuous for all and is continuous for all  is continuous on a set by using Theorem 5 in Section 13.2.

12.   Since is continuous for all and is continuous for all  is continuous on a set by using Theorem 5 in Section 13.2.  