1. LIMIT AND CONTINUITY
(NPD)

2. 3.1 Limit Function at One Point
An intuitive
Let
This function is not defined at x = 1 ( dividing by 0).
We still ask, what happening to f(x) as x approaches 1.
The following table shows some values of f(x) for x near 1 x
0.9
0.99
0.999
0.9999 1
1.0001
1.001
1.01
1.1
f(x)
1.9
1.99
1.999
1.9999 2
2.0001
2.001
2.01
2.1

3. Geometrically 1
The result in the table and graphic suggest that the values of f(x) approaches 2 as x approaches 1.
f(x) 2
We denote this by writing
f(x) x x
we read : the limit as x approaches 1 of is 2
Definition (intuitive) :
Means that when x is near but different from c, then f(x) is near L.

5. Example 1: a. b. c. d.
If values of x approach 0 we get value of f(x): x 1 -1 1 -1 ?
If x approaches 0, sin(1/x) does not approach to any number.
Thus limit does not exist.

9. Right- and Left-Hand Limits (one-sided limits)
If x approaches c from the left side (from number on which is smaller than c), we get the left-hand limit, denoted by x c

10. Right- and Left-Hand Limits (one-sided limits)
If x approaches c from the right side (from number on which is greater than c), we get the right-hand limit, denoted by c x

11. Example 4

12. Example 5
Let
a. Evaluate
b. Evaluate
c. Evaluate
d. Sketch graph of f(x)

13. Answer
Because the formula of f(x) change at x = 0, then we find
the one-sided limit at point x = 0.
b. Because the formula of f(x) change at x = 1, then we find
the one-sided limit at point x = 1.
c. The formula of f(x) does not change at x = 2, thus

14. d. 3
at x = 1 limit does not exist 1
For x 0
For 0 < x < 1
For x
f(x) = x
Graph: parabola
Graph: straight line
Graph: parabola

15. 2. Find the value of c so that
has a limit at x = -1
Solution :
f(x) has a limit at x = -1 if the limit from the right = the limit from the left
limit exist if
3 + c = 1 - c
C = -1

16. EXERCISES A. For the function f graphed below Find: 1. 4. f(-3) f(-1)2. 5. 6.
f(1) 3.

17. B.
1. Let
Evaluate
b. Does exist? if it limit exist evaluate
2. Let
, evaluate a. b. c.
3. Let
, evaluate a. b. c.

18. Theorem
Let
then

19. The Squeeze Theorem (Sandwich)
Let for all x in some open interval containing the point c and
then
Example 6: Evaluate
Because
and
then

24. Example 7 :
x  0 equivalent with 4x  0

25. Exercises
Evaluate: 1. 2. 3. 4.

26. The Precise Definition of a Limit

28. In general

31. Infinite Limits and Limits at Infinity
a. Infinite Limits
Remark :
g(x)  0 from upward mean g(x) approaches 0 from positive value of g(x).
g(x)  0 from downward mean g(x) approaches 0 from negative value of g(x).

32. Example 8: Evaluate a. b. c.
Solution a.
, g(x) = x - 1 approaches 0 from the downward
thus b.
, approaches 0 from the upward
thus

33. If x approaches from the right then sin(x) approaches 0 from downward
Because
f(x)=sinx
and x
If x approaches from the right then sin(x) approaches 0 from
downward
thus

34. Limits at Infinity

36. Example 9 : Evaluate
Solution
= 1/2

38. Example 10: Evaluate
Solution
= 0

39. Example 11 : Evaluate
Solution:
If x  , It is form ( )

40. Exercise
Evaluate 1. . . 2. 3. 4. 5. 6.

41. Continuity A Function f(x) is said continuous at x = c if
(i) f(c) is defined
(ii)
(iii)
If one or more of these 3 conditions fails, then f is discontinuous at c and c is a point of discontinuity of f
(i)
f(c) does not exist c
f is not continuous at c

42. (ii)
Left-hand limit ≠ right-hand limit ,
Thus limit does not exist at x = c c
f is not continuous at x = c
f(c) ●
(iii)
f(c) is defined L
exist
but c
f is not continuous at x = c

43. f(c) is defined
(iv)
exist
f(c) c
f(x) is continuous x=c
Removable discontinuity
For case (i) discontinuity can be removed by define f(c) = limit of function at c

44. Example
Determine whether f ,g ,and h are continuous at x = 2 a. b. c.
Solution:
a. f is not defined at x = 2 (0/0)
f is not continuous at x = 2
b. g(2) = 3,
g is not continuous at x=2

45. c.
h is continuous at x = 2

46. Continuity From the Left and the Right
A function f(x) is called continuous from the left at x =a if
A function f(x) is called continuous from the right at x = a if
f(x) is continuous at x = a if f(x) is continuous from the left and right
Example : Find the value of a so that f(x) will be continuous at x = 2

47. Solution :
f(x) is continuous at x = 2, if :
1. f is continuous from the left at x = 2
2 + a = 4a – 1
-3a = -3
a = 1
2. f is continuous from the right at x = 2
trivial

48. Problems 1. Let Determine whether f is continuous at x = -1
2. Find the value of a + 2b so that f(x) will be continuous everywhere
3. Find the values of a and b so that f(x) will be continuous at x = 2

49. Continuity on a Closed Interval
A function f(x) is said to be continuous on closed interval [a,b] if
1. f(x) is continuous on ( a,b )
2. f(x) is continuous from the right at x = a
3. f(x) is continuous from the left at x = b

50. Theorem A Polynomials are continuous function
A rational function is continuous everywhere except at the points where the denominator is zero
Theorem B
Let , then
f(x) is continuous everywhere if n is odd
f(x) is continuous for positive number if n is even
Example: In which x s.t is continuous ?
From theorem, f(x) is continuous for x – 4 > 0 or x > 4.
Thus f(x) is continuous at [4, )
f(x) is continuous from the right at x = 4

53. Example
Show that there is a root of the equation 𝑥 3 −𝑥 −1=0 between 1 and 2

54. Problems
A. Find the points of discontinuity, if any 1. 3. 2.
B. Find the points where f(x) are continuous 1. 2.