1. Intuitive Concepts of Limits, Continuity and Differentiability

2. Section 3 Limit of a Function
An introduction to limits from a numerical point of view

3. Intuitive Definition. Let y = f(x) be a function
Intuitive Definition. Let y = f(x) be a function. Suppose that a and L are numbers such that
whenever x is close to a but not equal to a, f(x) is close to L;
as x gets closer and closer to a but not equal to a, f(x) gets closer and closer to L; and
suppose that f(x) can be made as close as we want to L by making x close to a but not equal to a.
Then we say that the limit of f(x) as x approaches a is L and we write

4. Numerical and Graphical Exploration 1
After working through these materials, the student should be able
to obtain numerical evidence for the calculation of limits;
to determine what appears to be the limit from the numerical evidence; and
to become aware of some of the problems in using numerical evidence for the calculation of limits.

5. Numerical and Graphical Exploration 2
Don’t trust your calculators!
Numerical and Graphical Exploration 2
After working through these materials, the student should be able
to obtain numerical evidence for the calculation of limits;
to determine what appears to be the limit from the numerical evidence; and
to become aware of some of the problems in using numerical evidence for the calculation of limits.

6. Numerical and Graphical Exploration 3
Read p.11
Numerical and Graphical Exploration 3
After working through these materials, the student should be able
to obtain numerical evidence for the calculation of limits;
to determine what appears to be the limit from the numerical evidence; and
to become aware of some of the problems in using numerical evidence for the calculation of limits.
Need for considering both right-hand and left-hand limits.

7. Find Limits by graphs

8. Example 3.1
Discuss and
sketch the graph of

9. Section 4 Infinity
Type 1
A function f(x) tends to positive infinity (respectively negative infinity) as x tends to a if f(x) increases ( respectively decreases) without bounds as x tends to a in any manner.
In symbol, we write limx→af(x)=+∞
(respectively limx→af(x)= -∞)

10. Example 1

11. Example 2

12. Example 3

13. Type 2 A function f(x) approaches to L as x approaches to + (or - )
In symbol,

14. Example 4

15. Type 3
f(x) approaches to a limit L as the absolute value of x increases without bounds.
In symbol,

16. Example 5

17. Example 6

18. By intuitive concept,

19. On the other hand,

20. Section 5 Evaluation of Limits
Limit Theorems

21. Examples

22. Section 6 Continuity of a Function
Definition 6.1
A function f is continuous at x = a if and only if

23. Discussion on Ex.1.6 Q.1

24. 1.Exercise on Continuity
2.Exercise on the Continuity of piecewise defined functions
(with animation)

25. Illustrative Examples
6.1 State the points of discontinuity of the following functions:
(a) f(x) = tanx
(b) g(x) = x – [x]
(c) h(x) = (x2 – 4)/(x+2)
x=(2n-1)π/2, nεZ
xεZ
X= -2

26. Example 6.2 Is the function continuous at x=0?
If not, what should be f(0) so that f(x) is continuous there.
Solution:
∵limx→0f(x)=limx→0sinx/x = 1 and
f(0) = 0≠1
∴f(x) is not continuous at x = 0.
In order that f(x) is continuous at x = 0,
f(0) = 1.
Ex.1.6, Q.2,3