Ya

1. BAB 5
Media Pembelajaran Matematika
Limit Fungsi Aljabar

2. Sub Bab
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of the section here
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of the section here
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01
02
03
04
Definisi Limit Teorema Substitusi
Limit Satu Sisi Teorema Apit
Teorema Limit Kontinuitas Fungsi
05
06 You can describe the topic
of the section here
You can describe the topic
of the section here

3. Definisi Limit
01
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4. Sejauh ini kita telah memahami pengertian dan definisi limit secara intuitif
(perasaan) dengan menggunakan definisi sementara : "jika 𝑥 mendekati 𝑐 maka fungsi
𝑓(𝑥) akan mendekati 𝐿". Definisi sementara ini, telah memberi kemudahan dalam
memahami pengertian dan menghitung nilai limit fungsi dengan empat cara yang telah
dibahas. Namun demikian, kalimat: "jika 𝑥 mendekati 𝑐 maka fungsi 𝑓(𝑥) akan
mendekati 𝐿" adalah "definisi yang tidak tegas" secara matematika.
Pada abad ke-19, matematikawan Augustin-Louis Cauchy (1789 - 1857) dan Karl
Weierstrass (1815 - 1897) memperjelas gagasan tentang limit dan membangun definisi
yang paling tepat tentang limit.
Definisi Limit

5. Pernyataan tentang limit
lim
𝑥→𝑐
𝑓 𝑥 = 𝐿
Bermakna bahwa untuk setiap ε > 0 yang diberikan (berapa pun kecilnya), terdapat
bilangan lain yang sepadan yakni δ > 0 sedemikian rupa sehingga
𝑓 𝑥 − 𝐿 < 𝜀 bilamana 0 < 𝑥 − 𝑐 < 𝛿 ; yakni,
0 < 𝑥 − 𝑐 < 𝛿 ⟹ 𝑓 𝑥 − 𝐿 < 𝜀
Definisi Limit

6. Limit Satu Sisi
02
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7. Ketika suatu fungsi mempunyai lompatan (seperti halnya 𝑥 pada setiap bilangan
bulat), maka limit tidak ada pada setiap lompatan. Fungsi-fungsi yang demikian
menyarankan perkenalan tentang limit-limit satu sisi (one side limits). Misalkan
lambang 𝑥 → 𝑐+
bermakna bahwa 𝑥 mendekati 𝑐 dari kanan, dan 𝑥 → 𝑐−
bermakna
bahwa 𝑥 mendekati 𝑐 dari kiri.
Definisi Limit Kiri dan Limit Kanan
Untuk mengatakan bahwa lim
𝑥→𝑐+
𝑓 𝑥 = 𝐿 berarti bahwa ketika 𝑥 dekat tetapi pada
sebelah kanan 𝑐, maka 𝑓(𝑥) dekat ke-𝐿. Dari sini 𝐿 kemudian disebut dengan nilai limit
kanan di 𝑥 = 𝑐. Demikian pula, Untuk mengatakan bahwa lim
𝑥→𝑐−
𝑓 𝑥 = 𝐿 berarti bahwa
ketika 𝑥 dekat tetapi pada sebelah kiri 𝑐, maka 𝑓(𝑥) dekat ke-𝐿. Dari sini 𝐿 kemudian
disebut dengan nilai limit kiri di 𝑥 = 𝑐.
Limit Satu Sisi

8. Jadi walaupun lim
𝑥→2
𝑥 adalah benar untuk menuliskan
Limit suatu fungsi 𝑓(𝑥) dikatakan ada dan nilainya adalah 𝐿 jika nilai limit
arah kiri fungsi itu sama dengan nilai limit arah kanannya. Jadi nilai limit
fungsi 𝑓(𝑥) ketika 𝑥 → 𝑐 adalah sama dengan 𝐿 jika dan hanya jika nilai limit
arah kiri fungsi tersebut sama dengan nilai limit arah kanannya.
Limit Satu Sisi
lim
𝑥→2
𝑥 = 1 dan lim
𝑥→2
𝑥 = 2
Teorema
lim
𝑥→𝑐
𝑓 𝑥 = 𝐿 jika dan hanya jika lim
𝑥→𝑐−
𝑓 𝑥 = 𝐿 dan lim
𝑥→𝑐+
𝑓 𝑥 = 𝐿

9. Mengatakan lim
𝑥→𝑐+
𝑓 𝑥 = 𝐿 berarti bahwa untuk setiap 𝜀 > 0, terdapat 𝛿 > 0 yang
berpadanan sedemikian rupa sehingga
Limit Satu Sisi
0 < 𝑥 − 𝑐 < 𝛿 ⟹ 𝑓 𝑥 + 𝐿 < 𝜀
Definisi Limit Kanan
Definisi Limit Kiri
Mengatakan lim
𝑥→𝑐−
𝑓 𝑥 = 𝐿 berarti bahwa untuk setiap 𝜀 < 0, terdapat 𝛿 < 0 yang
berpadanan sedemikian rupa sehingga
0 < 𝑐 − 𝑥 < 𝛿 ⟹ 𝑓 𝑥 − 𝐿 < 𝜀

10. Teorema
Limit 03
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11. Teorema Limit
Teorema A Teorema Limit Utama
Misalkan 𝑛 bilangan bulat positif, 𝑘 konstanta, serta 𝑓 dan 𝑔 adalah fungsi-fungsi yang
mempunyai limit di 𝑐. Maka
1. lim
𝑥→𝑐
𝑘 = 𝑘 ;
2. lim
𝑥→𝑐
𝑥 = 𝑐;
3. lim
𝑥→𝑐
𝑘𝑓 𝑥 = 𝑘 𝑙𝑖𝑚
𝑥→𝑐
𝑓 𝑥 ;
4. lim
𝑥→𝑐
𝑓 𝑥 + 𝑔 𝑥 = lim
𝑥→𝑐
𝑓 𝑥 + lim
𝑥→𝑐
𝑔 𝑥 ;
5. lim
𝑥→𝑐
𝑓 𝑥 − 𝑔 𝑥 = lim
𝑥→𝑐
𝑓 𝑥 − lim
𝑥→𝑐
𝑔 𝑥
6. lim
𝑥→𝑐
𝑓 𝑥 . 𝑔 𝑥 = lim
𝑥→𝑐
𝑓 𝑥 . lim
𝑥→𝑐
𝑔 𝑥 ;
7. lim
𝑥→𝑐
𝑓 𝑥
𝑔 𝑥
=
lim
𝑥→𝑐
𝑓 𝑥
lim
𝑥→𝑐
𝑔 𝑥
, asalkan lim
𝑥→𝑐
𝑔 𝑥 ≠ 0
8. lim
𝑥→𝑐
[𝑓 𝑥 ]𝑛
= lim
𝑥→𝑐
𝑓(𝑥)
𝑛
;
9. lim
𝑥→𝑐
𝑛
𝑓(𝑥) = 𝑛
lim
𝑥→𝑐
𝑓(𝑥) , asalkan lim
𝑥→𝑐
𝑓(𝑥) > 0 ketika 𝑛 genap.

12. Penerapan Teorema Limit Utama
1. Carilah lim
𝑥→3
2𝑥4
Penyelesaian
Teorema Limit
lim
𝑥→3
2𝑥4
= 2 lim
𝑥→3
𝑥4
= 2 lim
𝑥→3
𝑥
4
= 2 3 4
= 162
2. Carilah lim
𝑥→4
3𝑥2 − 2𝑥
Penyelesaian
lim
𝑥→4
3𝑥2
− 2𝑥 = lim
𝑥→4
3𝑥2
− lim
𝑥→4
2𝑥 = 3 lim
𝑥→4
𝑥2
− 2 lim
𝑥→4
𝑥
lim
𝑥→4
3𝑥2
− 2𝑥 = 3 lim
𝑥→4
𝑥
2
− 2 lim
𝑥→4
𝑥 = 3(4)2
− 2 4 = 40

13. Penerapan Teorema Limit Utama
3. Carilah lim
𝑥→4
𝑥2+9
𝑥
Penyelesaian
Teorema Limit
4. Jika lim
𝑥→3
𝑓(𝑥) = 4 dan lim
𝑥→3
𝑔(𝑥) = 8, carilah lim
𝑥→3
𝑓2
𝑥 . 3
𝑔(𝑥)
Penyelesaian
lim
𝑥→4
𝑥2+9
𝑥
= lim
𝑥→4
𝑥2+9
lim
𝑥→4
=
lim
𝑥→4
𝑥2+9
4
=
1
4
lim
𝑥→4
𝑥2 + lim
𝑥→4
9
lim
𝑥→4
𝑥2+9
𝑥
=
1
4
lim
𝑥→4
𝑥
2
+ 9 =
1
4
42 + 9 =
5
4
lim
𝑥→3
𝑓2
𝑥 . 3
𝑔(𝑥) = lim
𝑥→3
𝑓2
𝑥 . lim
𝑥→3
3
𝑔(𝑥)
lim
𝑥→3
𝑓2
𝑥 . 3
𝑔(𝑥) = lim
𝑥→3
𝑓(𝑥)
2
. 3
lim
𝑥→3
𝑔(𝑥)
lim
𝑥→3
𝑓2
𝑥 . 3
𝑔(𝑥) = [4]2
.
3
8 = 32

14. Teorema Substitusi
04
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15. Ketika kita menerapkan Teorema B, kita katakan kita menghitung limit dengan substitusi.
Tidak semua limit dapat dihitung dengan substitusi; tinjau lim
𝑥→1
𝑥2−1
𝑥−1
. Teorema substitusi
tidak diterapkan disini karena penyebut adalah 0 ketika 𝑥 = 1, tetapi limit memang ada.
Teorema B Teorema Substitusi
Jika 𝑓 fungsi rasional atau fungsi polinomial, maka
lim
𝑥→𝑐
𝑓 𝑥 = 𝑓(𝑐)
Asalkan 𝑓 𝑐 terdefinisi. Dalam kasus fungsi rasional, ini bermakna bahwa nilai penyebut pada 𝑐 tidak nol.
Teorema Substitusi
Perhitungan Limit “dengan Substitusi”

16. Teorema Apit
05
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17. Penyelesaian
Teorema D Teorema Apit (Sequeeze Teorem)
Misalkan 𝑓, 𝑔, dan ℎ adalah fungsi yang memenuhi 𝑓(𝑥) ≤ 𝑔(𝑥) ≤ ℎ(𝑥) untuk semua 𝑥 dekat 𝑐,
terkecuali mungkin pada 𝑐. Jika lim
𝑥→𝑐
𝑓 𝑥 = lim
𝑥→𝑐
ℎ 𝑥 = 𝐿 maka lim
𝑥→𝑐
𝑔 𝑥 = 𝐿
Teorema Apit
Asumsikan bahwa kita telah membuktikan
1 − 𝑥2
6
≤
(sin 𝑥)
𝑥
≤ 1 untuk semua 𝑥 yang dekat tetapi
berlainan dengan 0. Apa yang kita simpulkan tentang lim
𝑥→𝑐
sin 𝑥
𝑥
= 1
Contoh
Misalkan 𝑓 𝑥 =
1 − 𝑥2
6
, 𝑔 𝑥 =
(sin 𝑥)
𝑥
, ℎ 𝑥 = 1 . menyusul bahwa lim
𝑥→0
𝑓 𝑥 = 1 = lim
𝑥→0
ℎ(𝑥)
dan akibatnya, menurut Teorema C
lim
𝑥→0
sin 𝑥
𝑥
= 1

18. Kontinuitas
Fungsi 06
You can enter a subtitle
here if you need it

19. Sebuah fungsi dikatakan kontinu dalam selang 𝑎 ≤ 𝑥 ≤ 𝑏 jika grafik fungsi tersebut
tersambung utuh, tidak berputus, dan tidak memiliki titik diskontinu di dalam selang
tersebut.
Kontinuitas Fungsi
Kontinuitas
Jika 𝑥 = 𝑐 adalah sebuah titik yang berada di dalam selang 𝑎 ≤ 𝑥 ≤ 𝑏, maka sebuah fungsi
𝑓(𝑥) dikatakan kontinu di titik 𝑐 jika memenuhi ketiga syarat berikut ini.
a. Nilai 𝑓(𝑐) terdefinisi (ada);
b. lim
𝑥→𝑐
𝑓(𝑥) ada;
c. lim
𝑥→𝑐
𝑓 𝑥 = 𝑓(𝑐)
Jika salah satu dari ketiga syarat ini tak terpenuhi, maka fungsi 𝑓(𝑥) dikatakan tidak kontinu
dititik 𝑐. Jika 𝑓(𝑥) tidak kontinu dititik 𝑐, maka 𝑓(𝑥) dikatakan diskontinu di 𝑐

20. Secara geometri, gambar dibawah ini menampilkan tiga keadaan dimana fungsi 𝑓(𝑥) tidak
kontinu (diskontinu) di titik 𝑥 = 𝑐, sebagai lawan (kebalikan) dari syarat kontinuitas yang
disebutkan di atas.
Kontinuitas Fungsi

21. Contents of this template
Secara geometri, gambar dibawah ini menampilkan tiga keadaan dimana fungsi 𝑓(𝑥) tidak
kontinu (diskontinu) di titik 𝑥 = 𝑐, sebagai lawan (kebalikan) dari syarat kontinuitas yang
disebutkan di atas.
Kontinuitas Fungsi

22. Terima Kasih
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23. Vector basics: introduction and fundamentals
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clear? Lists like this one:
● They’re simple
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● You’ll never forget to buy milk!
And the most important thing: the audience won’t
miss the point of your presentation

24. Components
Vector fundamentals
Venus has a beautiful name and
is the second planet from the
Sun. It’s hot and has a very
poisonous atmosphere
Mercury is the closest planet to
the Sun and the smallest one in
the Solar System—it’s only a bit
larger than the Moon
Definition

25. Required knowledge
Mercury is the closest
planet to the Sun and the
smallest of them all
Venus has a beautiful
name and is the second
planet from the Sun
Despite being red, Mars is
actually a cold place. It’s
full of iron oxide dust
Algebraic Functions Graphing

26. Four concepts
Venus has a beautiful name and is the
second planet from the Sun. It’s
terribly hot, even hotter than Mercury
Earth is the third planet from the Sun
and the only one that harbors life in
the Solar System
Despite being red, Mars is actually a
cold place. It’s full of iron oxide dust,
which gives the planet its reddish cast
Jupiter is the biggest planet in the
Solar System. It’s the fourth-brightest
object in the night sky
Geometry
Trigonometry
Exponents
Logarithms

27. Venus has a beautiful name,
but also high temperatures
Neptune is the fourth-largest
planet in the Solar System
Reviewing concepts
Despite being red, Mars is
actually a very cold place
Earth is the third planet from
the Sun and has life
Saturn is the second-largest
planet in the Solar System
Jupiter is a gas giant and has
around eighty moons
Graphing Exponential Polynomial
Rational Sequences Series

28. Awesome
words

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31. A picture is
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thousand words

32. 98,300,000
Big numbers catch your audience’s attention

33. Jupiter’s rotation period
9h 55m 23s
333,000
The Sun’s mass compared to Earth’s
386,000 km
Distance between Earth and the Moon

34. Mercury is the closest
planet to the Sun and the
smallest of them all
Category A
Venus has a beautiful
name and is the second
planet from the Sun
Category B
Despite being red, Mars is
actually a cold place. It’s
full of iron oxide dust
Category C
Percentage breakdown
50% 75%
25%

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38. Venus
Geographical vector analysis
Venus is the second planet
from the Sun. It’s terribly hot,
and its atmosphere is
extremely poisonous. It’s the
second-brightest natural
object in the night sky after
the Moon

39. Lesson timeline
Venus
Venus is the second
planet from the Sun
Mercury
Mercury is the closest
planet to the Sun
Mars
Despite being red,
Mars is a cold place
Jupiter
Jupiter is the biggest
planet of them all
1st semester
Section 1 Section 2 Section 3 Section 4
2nd semester

40. Key concepts
Pre-calculus
By studying pre-calculus,
students develop critical
thinking skills, logical reasoning,
and the ability to analyze and
interpret mathematical models
Functions & graphs
Define functions and emphasize their importance
Trigonometry
Introduce trigonometric ratios and their applications
Logarithmic functions
Explain the properties and applications
Rational functions
Rational functions, emphasizing simplifying and solving
Equations and inequalities
The systems of linear and nonlinear equations

41. Pre-calculus lesson plan
Lesson no. Topic Key concepts
01 Introduction to pre-calculus Definition and importance
02 Functions and graph Domarin, range, graphing
03 Trigonometry Ratios, functions, identities
04 Polynomial and rational functions Factoring, simplifying, rational
05 Exponential and logarithmic Properties, equations, graphing
06 Systems of equations Applications, solving systems

42. Vector options distribution
Follow the link in the graph to modify its data and then paste the new one here. For more info, click here
Magnitude
Mercury is quite a
small planet
Direction
Jupiter is an
enormous planet
Scalar
Venus has very
high temperatures
Cross
Saturn is a gas
giant with rings

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44. Some tips
Here are some tips for solving equations in high school:
Properties
Mercury is the closest
planet to the Sun
Check
Jupiter is the biggest planet
in the Solar System
Order
Despite being red, Mars is
actually a cold place
Combine
Neptune is the farthest
planet from the Sun
Isolate
Venus is the second planet
from the Sun
Attention
Saturn has a high number
of moons, like Jupiter

45. Introduction to the
exercises
You can give a brief description of the topic you
want to talk about here. For example, if you want to
talk about Mercury, you can say that it’s the
smallest planet in the entire Solar System

46. Graph of y = sin x
The following table presents the values
of the sine function (sin x) for angles
ranging from 0 to 360 degrees
You can represent these values visually by
creating a graph. Here is an illustration of how
the graph would look like
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,4
1,2
1,0
0,8
0,6
0,4
0,2 30 60 90 120 150 180 210 240 270 310 330 360
X Y
0 0
30 0,5
60 0,8
90 1,0
120 0,8
150 0,5
180 0
210 -0,5
240 -0,8
270 -1,0
300 -0,8
330 -0,5
360 0

47. Identifying functions
To determine whether each table of values represents a function, we need to check if
there is a unique output (y-value) for every input (x-value) in the table. If there is no
repetition of x-values and each x-value corresponds to a single y-value, then the table
represents a function. State whether each table of values represents a function
X Y
-12 2
-10 10
0 -2
5 -6
8 -11
15 -15
X Y
9 -18
-20 0
-6 1
-17 16
9 17
11 19
X Y
4 -20
1 -17
4 -14
16 5
10 0
-19 -16
X Y
-15 18
-11 18
-14 18
-9 18
-1 18
-5 18

48. Some equations
Determine the relationship between the equations. Place
greater than (>), less than (<) or equal to (=) in the space provided
Where x=3
a) 5x + 4 ____ 3x + 15
b) x + 23 ____ 5x - 4
c) 7x - 2 ____ 4x + 4
d) 2x + x ____ 6x - 5
e) 6x + 2 ____ 4x + 4
f) 3x + 5 ____ 6x - 4
Where x=7
a) 3x - x ____ 4x + 14
b) 2x + 12 ____ 3x - 4
c) x + x + 7 ____ 5x
d) 2x + 10 ____ 5x - 5
e) 6x - 18 ____ 4x - 4
f) 8x ____ 3x + 2x + 15

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