13.2 Limits and Continuity day 1

1. 13.2 Limits and Continuity day 1

2. Example 2 Evaluate: Discuss any points of discontinuity.

4. Example 4 Show the limit does not exist:

5. Example 4

6. Example 3 Evaluate:

7. Evaluate: To solve this problem, first substitute to find that it is not continuous. To find out if the limit exists try approaching (0,0) on the lines: x=0, y=0, y=x and y=-x (next slide) Example 3

8. Evaluate: If all approaching from all these directions equals the same value then we suspect that the limit exists. We can gather further evidence by looking that the graph. However, proving that it exists can be challenging. (not on the tests but explained in the next slides) Example 3

9. A non AP topic that was not Taught in BC • Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function f as above and an element c of the domain I, f is said to be continuous at the point c if the following holds: For any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain of f with c − δ < x < c + δ, the value of f(x) satisfies • Alternatively written, continuity of f : I → R at c ∈ I means that for every ε > 0 there exists a δ > 0 such that for all x ∈ I,: • More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(c), we simply need to choose a small enough neighborhood for the x values around c, and we can do that no matter how small the f(x) neighborhood is; f is then continuous at c. • In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.

10. delta neighborhood(with an open disc)(don’t panic this is not a major part of the homework nor on the tests) A delta neighborhood is the set of all of the points less than some distance delta from a starting point. This is denoted algebraically by:

11. This consists of all of the points that are on the boundary and the interior of the region R. A delta neighborhood (with a closed disc)

12. If you approach the point (a,b) from any direction you must approach the same z value.

13. Note:The definition of the limit of a function of two (independent) variables is similar to the limit of a function of a single variable. However, to determine if a function of a single variable has a limit you need only check from two directions (the left and right). If the function approaches the same value from those two directions you can conclude that the limit exists. With a limit of two variables (x,y) is allowed to approach (a,b) from any direction.

14. Note: this is essentially the same definition that was used for 1 independent variable. Definition of Continuity of a Function of Two Variables

15. Q: How does one insult a mathematician? A: You say: “your brain is smaller than any >0!"

16. delta neighborhood(with an open disc)(don’t panic this is not a major part of the homework nor on the tests) A delta neighborhood is the set of all of the points less than some distance delta from a starting point. This is denoted algebraically by: