PPP Limit dan Continue

1. Limit and continuity
GEORGE SIMANJUNTAK
RIZKA HUTASOIT – 4213141049
SUANNA SIPAYUNG
TIARMA MATONDANG
GRUP 3 :

2. Concep limit :
Definition of limit :
f(x)
x = 1, so f(x) = = undefinite
The function above is undefined if x = 1, but it is still declare the value of its approach if
x comes closer to 1. Illustrated in the following chart :
limit
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 →?
←
2,0001 2,001 2,01 2,1
When viewed from left : When viewed from right:
From the chart above, it appears that the f(x) is approaching 2, if x = 1. Systematically written:

4. ● F(x) =
If x = 1, so F(1) = = =
When viewed from left: When viewed from right :
∞ ∞
● So it could be concluded that the function of the above limit is :
● That left limit ≠ right limit, so :
x 0,9 0,99 0,9999 1 1,0001 1,001 1,01 1,1
F(x) -10 -100 -10000 10000 1000 100 10
lim
𝑥 →1
1
x −1
= h
𝑁𝑜𝑡 𝑖𝑛𝑔
:

5. DEFINITION OF LIMIT
ε > 0, ∃ δ > 0 such that | x – c | < δ => |
f(x) – L | < ε

6. LIMIT KIRI DAN LIMIT KANAN

7. LIMIT FUNCTION PROPERTIES
● k,c is constana
● exist
● exist
● n is integer
Then apply :
● = ,
● = [
● = , dengan f(x) 0 jika n genap
≥

8. APIT PRICIPLE
Ex, f(x) g(x) h(x) for x around c and and , so
≤ ≤

9. Example :
● Count ( x – 1 )² sin =
The value of range sine is -1 to 1
-1 sin 1
≤ ≤
So, – ( x – 1 )² ( x – 1 )² sin ( x – 1 )²
≤ ≤
Because lim – ( x – 1 )² = 0 and lim ( x – 1 )² = 0
So, according to the apit principle obtained that ;
( x – 1 )² sin = 0

10. Limit of trigonometric functions :
How to finish ?
● Substitution.
● If undefinite form, change the function that make a undefinite value with
basic characteristic trigonometric functions.
The basic characteristics of trigonometric functions : The basic characteristics of trigonometric function if coefficient x
not 1
● = 1 1. =
● = 1 2. =
● = 1 =
● = 1 4. =

11. Example
● =
If substitutioned will produce undefinite form, sthe function that make
undefinite must changed with basic characteristics the limit of trigonometric
functions.
= 1
=
= (Use limit properties )
=
= 1. =

12. The basic characteristics of limit trigonometric functions if numerator and denominator is a trigonometric functions. :
=
=
=
=
● Example :
=
=
= )
= .
= 1. 5 = 5

13. ● The patterns of limit trigonometric function

14. Example
=
Triginimetric Identity:
1 - cos² x = sin² x
so, =
= .
= . =

15. Limit infinity and limit at infinity
● Limit infinity
f(x) = L ≠ 0 dan g(x) = 0
So, =
● +∞, if L>0 and g(x) 0 from above ( from the positive direction )
● - ∞ , if L>0 and g(x) 0 from below (from the negative direction)
● +∞, if L<0 and g(x) 0 from below (from the negative direction)
● - ∞, if L<0 and g(x) 0 from above (from the positive direction)
g(x) 0 from above means g(x) going to 0 from value g(x) positive
g(x) 0 from below means g(x) going to 0 from value g(x) negative

16. Example
Answer :
f(x) = , 1² + 1 = 2, 2 > 0 ( positive )
g(x) = x – 1
from left direction will going to 0 from below because x 1 from left means x
< 1, as a result x – 1 negative value
So because > 0 ( positive ) and g(x) = x -1 going to 0 from below ( negative )
so,
= - ∞

17. Limit at infinite
● such that x > M => | f(x) – L | <
Or if f(x) approaching L if x going to positive infinite.

18. Example
● =
Answer :
=

19. ● such that x > M => | f(x) – L | <
Or f(x) approaching L if x going to negative infinite.

20. Example
● =
= 0

21. Continuity
The function f(x) is said to be continuous at a point x = a, if:
Function f(x) said continu at the point x = a if :
• f(a) exist
• exist

22. If, any of the above conditions are not met, then f(x) is said to
be continuous at x = a.
So... these three conditions must be met, so that f(x) can be
said to be continuous at x = a. To clarify about continuity we
will look at and discuss the following examples...:D

23. EXAMPLES :
Is f(x) Continu at x = a?
● f(a) nothing because at x = a empty circle
So,
● f(x) not continu at x = a
function is unknown. So we cannot determine that the
value of the function is equal to the value of the limit

24. Is f(x) Continu at x = a?
● F(a) = L2 ( f(a) exist)
Left limit ≠ right limit
● = nothing

25. Is f(x) Continu at x = a?
● F(a) = L2 ( f(a) exist)
So,
● f(a) = f(x)
so, we can conclude that f(x) is
continuous at x = a

26. Erased Discontinuity

27. the meaning of the picture :]
So, the conclusion from the picture above is... the discontinuity that was
removed earlier by defining the value of the function at that point is the same
as the limit of the function.
Known, that
so it can be determined the f(a) = L
f(x) is continuous at x = a

28. More examples...!!
(example of a function)
If the first or earlier question uses a graph,now we will move on to questions
that are in the form of functions.So the question is....
● check if the following function is continuous at x = 2.
If not, state the reason

29. “Because the problem has changed a little....then, the conditions have also
changed slightly.”
To this :
Condition f(x) is continuous at x = 2 :
• f(a) exist
• exist

30. a. f(x) =
answer: function is not defined at x = 2
( returns the form )
it can be concluded that the first condition is wrong, so f (x) does not
allow continuous with x = 2

31. b. f(x) =
Answer :
● f(2) = 3
● = = = 4
● f(x) f(2)
4 3
So, f(x) not continuous at x = 2

32. c. f(x) =
Answer :
● f(2) = - 1 = 3
● = = 3
● = = 3
So, = 3
● = f(2)
So, the f(x) is continous x = 2

33. continuous right and continuous left
The function f(x) is called left continuous at x = a, if
f(a)
The function f(x) is called right continuous at x = a, if
so, the function f(x) is continuous at x = a , if it is right continuous and left
continuous at x = a

34. Examples :
determine the constant a so that the function
f(x) =
Answer :
the function f(x) is continuous at x = 2 , if it is continuous left and right continuous at x =
2
1. check that f(x) is continuous at x = 2