Math 311-01/02

2-20-2018

* [Limits and continuity problems]*

1. Compute .

I can compute the limit by plugging in. (This is another way of saying that is continuous at .) Thus.

2. Compute .

I can compute the limit by plugging in.

3. Compute .

Hint: Try letting along the line and along the line .

Substituting yields the indeterminate form .

Set and let . I have

Set and let . I have

Since approaches different numbers depending on how approaches , the limit is undefined.

You can see the different "levels" being approached near the origin in the graph of the surface:

4. Compute .

Hint: Try letting along the line and along the parabola .

If you approach along the line , you get

If you approach along the parabola , you get

Since approaches different numbers depending on how approaches , the limit is undefined.

You can see the different "levels" being approached near the origin in the graph of the surface:

5. Compute by converting to polar coordinates.

Let and . Then

Then

6. Compute .

Hint: Try letting along the line , and along the line .

If you approach along the line , , you get

If you approach along the line , you get

Since you get different limits by approaching in different ways, is undefined.

7. A function is defined by

Determine whether f is continuous at .

However, .

Since , the function is not continuous at .

*Two wrongs don't make a right, but they make a good excuse.*
- *Thomas Szasz*

Copyright 2018 by Bruce Ikenaga