Limits and Continuity in the Plane

Now there are infinitely many ways to approach.

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

What this lesson teaches

The student decides whether a limit of two variables exists using the path test.

A point in the plane can be approached along infinitely many paths, and all must agree.
Two paths giving different values prove the limit does not exist.
Agreeing paths prove nothing, since another path may still differ.

Straightforward practice. Get the method working first.

5 problems

Two paths give 2 and 5. Does the limit exist? 1 yes, 0 no.
Answer 0
Why No.
What can the path test actually prove?
Answer That a limit does not exist, by finding two paths that disagree.
Why It disproves; it cannot prove.
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 3
Why Infinitely many directions exist.
f(x, y) = 3x + y². What is the limit at (1, 2)?
Answer 7
Why A polynomial is continuous, so substitute.
xy/(x² + y²) along y = 0. What value does it give?
Answer 0
Why The numerator is zero.

The same ideas with more to keep track of.

3 problems

xy/(x² + y²) along y = x. What value does it give, as a decimal?
Answer 0.5
Why x² over 2x².
Does finding fifty agreeing paths prove a limit exists? 1 yes, 0 no.
Answer 0
Why The fifty-first might differ.
f(x, y) = x² − y² . What is the limit at (3, 2)?
Answer 5
Why 9 − 4.

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.
Answer Settles the question: Two paths give different values, The function is a polynomial · Settles nothing: Three paths give the same value, The x-axis path gives 0
Why A disagreement is conclusive; an agreement is not.
f is continuous at (a, b) and f(a, b) = 12. What is the limit there?
Answer 12
Why That is what continuity says.
The Two Paths: Two paths to a point give limits 3 and 7. Does the limit exist? 1 yes, 0 no.
Answer 0
Why No, it does not exist.
The Polynomial: f(x, y) = x² + 3y is continuous everywhere. What is the limit as (x, y) approaches (2, 1)?
Answer 7
Why 4 + 3 = 7.