This document discusses limits and continuity for functions of two variables. It defines a limit of a function f(x,y) as (x,y) approaches (a,b) as making the values of f(x,y) as close to L as desired by taking the point (x,y) sufficiently close to (a,b). It states the rules of limits can be added, subtracted, multiplied, composed and divided if the denominator's limit does not equal zero. A function is continuous at (a,b) if the limit exists as (x,y) approaches (a,b) and equals the function's value at (a,b). Examples are provided to demonstrate limits that do and do

1. LIMIT AND
CONTINUITY

2. Content

3. Introduction
• To develop calculus for functions of one variable, we needed to make sense of the concept of a limit, which we needed to
understand continuous functions and to define the derivative. Limits involving functions of two variables can be
considerably more difficult to deal with; fortunately, most of the functions we encounter are fairly easy to understand.
• Continuity what it means for a function of one variable to be continuous. In brief, it meant that the graph of the function did
not have breaks, holes, jumps, etc. We define continuity for functions of two variables in a similar way as we did for
functions of one variable.

4. LIMIT

5. Definition:
We write
lim(x,y)→(a,b) f (x,y) = L
and we say that the limit of f (x,y) as (x,y) approaches (a,b) is L if we can
make the values of f (x,y) as close to L as we like by taking the point (x,y) to
be sufficiently close to (a,b).

6. Easy Limits
• lim (x, y)→(a, b) x = a
• lim (x, y)→(a, b) y = b
• lim (x, y)→(a, b) c = c

7. Rule of Limits
Like regular limits , limits of multivariate functions can be
1. added
2. subtracted
3. multiplied
4. composed
5. divided ,provided the limit of the denominator is not zero

8. Limit of a
Polynomial

9. Limit of a
Rational
Function

10. Showing a limit does not exist

11. Example 1

12. Example 2

13. Example 3

14. Example 4

15. Example 5

16. Showing a limit does exists

17. CONTINUITY

18. Definition
A function f of two variables is said to be continuous at (a, b) if
(1). f(x, y) is defined at (a, b)
(2). lim (x, y)→(a, b) f (x, y) exists
(3). lim (x, y)→(a, b) f (x, y) = f(a, b)

19. Properties

20. Testing for Continuity

21. Example 2

22. Example 3

23. Example 4

24. Example 5

25. References
http://www.du.ac.in/index.php?page=virtual-learning

26. THANK YOU