Limits and Continuity in the Plane

Now there are infinitely many ways to approach.

12 problems · about 23 minutes · F-IF.B.4

NameDate

Figure — use these to answer the problems

The contours pull apart in different directions near the centre. Different paths to the same point can arrive at different heights.

Figure 1

Straightforward practice. Get the method working first.

5 problems

Two paths give 2 and 5. Does the limit exist? 1 yes, 0 no.
Answer
2
What can the path test actually prove?
- That a limit does not exist, by finding two paths that disagree.
- That a limit exists, by checking enough paths.
How many paths must agree for a two-variable limit to exist? Enter 1 for two, 2 for four, 3 for all of them.
Answer
f(x, y) = 3x + y². What is the limit at (1, 2)?
Answer
xy/(x² + y²) along y = 0. What value does it give?
Answer

The same ideas with more to keep track of.

3 problems

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each statement by whether it settles a two-variable limit.
Write each item under the heading it belongs to: Two paths give different values · Three paths give the same value · The function is a polynomial · The x-axis path gives 0
Settles the question
Settles nothing
f is continuous at (a, b) and f(a, b) = 12. What is the limit there?
Answer
The Two Paths
Two paths to a point give limits 3 and 7. Does the limit exist? 1 yes, 0 no.
Answer
The Polynomial
f(x, y) = x² + 3y is continuous everywhere. What is the limit as (x, y) approaches (2, 1)?
Answer