Flink Forward Berlin 2017: Dongwon Kim - Predictive Maintenance with Apache F...Flink Forward
SK telecom shares our experience of using Flink in building a solution for Predictive Maintenance (PdM). Our PdM solution named metatron PdM consists of (1) a Deep Neural Network (DNN)-based prediction model for precise prediction, and (2) a Flink-based runtime system which applies the model to a sliding window on sensor data streams. Efficient handling of multi-sensor streaming data for real-time prediction of equipment condition is a critical component of our product. In this talk, we first show why we choose Flink as a core engine for our streaming use case in which we generate real-time predictions using DNNs trained with Keras on top of TensorFlow and Theano. In addition, we present a comparative study of methods to exploit learning models on JVM such as directly using Python libraries on CPython embedded in JVM, using TensorFlow Java API (including Flink TensorFlow), and making RPC calls to TensorFlow Serving. We then explain how we implement the runtime system using Flink DataStream API, especially with event time, various window mechanisms, timestamp and watermark, custom source and sink, and checkpointing. Lastly, we present how we use the official Flink Docker image for solution delivery and the Flink metric system for monitoring and management of our solution. We hope our use case sets a good example of building a DNN-based streaming solution using Flink.
- Speaker: Servey Bochenkov - Head of Search @ TIKI
Search là một trong những feature quan trọng nhất đối với các website thương mại điện tử giúp khách hàng có thể dễ dàng tìm kiếm được sản phẩm mà mình mong muốn. Nhưng việc xây dựng một hệ thống search chất lượng nhưng vẫn đảm bảo tối ưu performance, resource sử dụng như RAM, CPU là một thách thức không hề nhỏ.
Đến với TechTalk #35 lần này, anh Sergey Bochenkov - với hơn 7 năm làm việc tại Cốc Cốc, hiện đang là Head of Search @ Tiki - sẽ chia sẻ cho chúng ta những ý tưởng cũng như khó khăn khi xây dựng language model dựa trên dữ liệu sản phẩm và search queries của Tiki cùng những dữ liệu khác được crawl từ các website để xây dựng Tiki spellchecker và autocorrection với một số nội dung nổi bật như:
- Quality optimizations idea
- Performance optimizations problems
- Giúp tăng 3-9% lượng mua hàng.
The Good, The Bad and The Ugly of Event Sourcing Dennis Doomen
In 2009, I first learned about Event Sourcing and Command Query Responsibility Seggregation (CQRS) at a training Greg Young gave in Utrecht, The Netherlands. I remember being awed by the scalability and architectural simplicity those styles provided. However, I also remembered the technical complexity that comes with it. In 2012, I was in charge of transitioning a CQRS-based system to Event Sourcing. I knew it would be non-trivial, but boy was I in for a surprise. So, over the last four years, I've experienced first-hand how a large group of developers had to deal with the transition. It's a brilliant solution for high-performance or complex business systems, but you need to be aware that this also introduces challenges most people don't tell you about. In this talk, I'd like to share you some of the most powerful benefits of ES, but also show you the flipside of the coin and cover some of the smaller and bigger challenges you'll run into it. Again, I love it and would apply it again without any doubt, but I really want you to understand the trade-offs before you jump on the Event Sourcing train.
PMM database open source monitoring solutionLior Altarescu
PMM - Percona Monitoring & Management is a open source monitoring and management tool that allows you to monitor many databases and other external services . It has a great insight for query and shows the explain plan in a visual way
Eventing Things - A Netflix Original! (Nitin Sharma, Netflix) Kafka Summit SF...confluent
Netflix Studio spent 8 Billion dollars on content in 2018. When the stakes are so high, it is paramount to track changes to the core studio metadata, spend on our content, forecasting and more to enable the business to make efficient and effective decisions. Embracing a Kappa architecture with Kafka enables us to build an enterprise grade message bus. By having event processing be the de-facto paved path for syncing core entities, it provides traceability and data quality verification as first class citizens for every change published.This talk will also get into the nuts and bolts of the eventing and stream processing paradigm and why it is the best fit for our use case, versus alternative architectures with similar benefits We will do a deep dive into the fascinating world of Netflix Studios and how eventing and stream processing are revolutionizing the world of movie productions and the production finance infrastructure.
This document provides an overview of Distributed Replicated Block Device (DRBD), which is a software-based storage solution that mirrors the contents of block devices across multiple nodes in real-time. It describes DRBD's architecture including kernel modules, user space tools, resource configuration, roles, modes, features around efficient synchronization and data verification, and limitations.
Flink Forward Berlin 2017: Dongwon Kim - Predictive Maintenance with Apache F...Flink Forward
SK telecom shares our experience of using Flink in building a solution for Predictive Maintenance (PdM). Our PdM solution named metatron PdM consists of (1) a Deep Neural Network (DNN)-based prediction model for precise prediction, and (2) a Flink-based runtime system which applies the model to a sliding window on sensor data streams. Efficient handling of multi-sensor streaming data for real-time prediction of equipment condition is a critical component of our product. In this talk, we first show why we choose Flink as a core engine for our streaming use case in which we generate real-time predictions using DNNs trained with Keras on top of TensorFlow and Theano. In addition, we present a comparative study of methods to exploit learning models on JVM such as directly using Python libraries on CPython embedded in JVM, using TensorFlow Java API (including Flink TensorFlow), and making RPC calls to TensorFlow Serving. We then explain how we implement the runtime system using Flink DataStream API, especially with event time, various window mechanisms, timestamp and watermark, custom source and sink, and checkpointing. Lastly, we present how we use the official Flink Docker image for solution delivery and the Flink metric system for monitoring and management of our solution. We hope our use case sets a good example of building a DNN-based streaming solution using Flink.
- Speaker: Servey Bochenkov - Head of Search @ TIKI
Search là một trong những feature quan trọng nhất đối với các website thương mại điện tử giúp khách hàng có thể dễ dàng tìm kiếm được sản phẩm mà mình mong muốn. Nhưng việc xây dựng một hệ thống search chất lượng nhưng vẫn đảm bảo tối ưu performance, resource sử dụng như RAM, CPU là một thách thức không hề nhỏ.
Đến với TechTalk #35 lần này, anh Sergey Bochenkov - với hơn 7 năm làm việc tại Cốc Cốc, hiện đang là Head of Search @ Tiki - sẽ chia sẻ cho chúng ta những ý tưởng cũng như khó khăn khi xây dựng language model dựa trên dữ liệu sản phẩm và search queries của Tiki cùng những dữ liệu khác được crawl từ các website để xây dựng Tiki spellchecker và autocorrection với một số nội dung nổi bật như:
- Quality optimizations idea
- Performance optimizations problems
- Giúp tăng 3-9% lượng mua hàng.
The Good, The Bad and The Ugly of Event Sourcing Dennis Doomen
In 2009, I first learned about Event Sourcing and Command Query Responsibility Seggregation (CQRS) at a training Greg Young gave in Utrecht, The Netherlands. I remember being awed by the scalability and architectural simplicity those styles provided. However, I also remembered the technical complexity that comes with it. In 2012, I was in charge of transitioning a CQRS-based system to Event Sourcing. I knew it would be non-trivial, but boy was I in for a surprise. So, over the last four years, I've experienced first-hand how a large group of developers had to deal with the transition. It's a brilliant solution for high-performance or complex business systems, but you need to be aware that this also introduces challenges most people don't tell you about. In this talk, I'd like to share you some of the most powerful benefits of ES, but also show you the flipside of the coin and cover some of the smaller and bigger challenges you'll run into it. Again, I love it and would apply it again without any doubt, but I really want you to understand the trade-offs before you jump on the Event Sourcing train.
PMM database open source monitoring solutionLior Altarescu
PMM - Percona Monitoring & Management is a open source monitoring and management tool that allows you to monitor many databases and other external services . It has a great insight for query and shows the explain plan in a visual way
Eventing Things - A Netflix Original! (Nitin Sharma, Netflix) Kafka Summit SF...confluent
Netflix Studio spent 8 Billion dollars on content in 2018. When the stakes are so high, it is paramount to track changes to the core studio metadata, spend on our content, forecasting and more to enable the business to make efficient and effective decisions. Embracing a Kappa architecture with Kafka enables us to build an enterprise grade message bus. By having event processing be the de-facto paved path for syncing core entities, it provides traceability and data quality verification as first class citizens for every change published.This talk will also get into the nuts and bolts of the eventing and stream processing paradigm and why it is the best fit for our use case, versus alternative architectures with similar benefits We will do a deep dive into the fascinating world of Netflix Studios and how eventing and stream processing are revolutionizing the world of movie productions and the production finance infrastructure.
This document provides an overview of Distributed Replicated Block Device (DRBD), which is a software-based storage solution that mirrors the contents of block devices across multiple nodes in real-time. It describes DRBD's architecture including kernel modules, user space tools, resource configuration, roles, modes, features around efficient synchronization and data verification, and limitations.
Stream Processing with Flink and Stream Sharingconfluent
Stream Processing with Apache Flink in the Cloud and Stream Sharing
Kai Waehner discusses stream processing use cases with Apache Flink and introduces Confluent's plans to offer a serverless Apache Flink service in Confluent Cloud. He also discusses Confluent Stream Sharing, which allows for easy and secure sharing of Kafka topics across organizations in real-time.
This document provides information about MongoDB replication and sharding. It discusses what replication is, how to set up replication on Windows including starting primary and secondary servers and verifying replication. It also discusses best practices for replication including always using replica sets, using replica sets to offload reads from primary, and using an odd number of replicas. The document also discusses how to set up MongoDB replication on Linux in a step-by-step process and how to check the replication status. It provides commands for adding and removing MongoDB instances from a replica set and making a primary secondary. Finally, it discusses what sharding is in MongoDB, the concept of sharding keys, and provides a high-level overview of implementing sharding in MongoDB including using
The slides of my talk about Vertical vs Horizontal Software Architecture at Equal Expert's Expert Talks Pune on 2019-09-14. It describes why vertical architectures are better than horizontal architectures for the perspective of change and architecture principles, and how this should reflect in the structure of source code.
- Replica sets in MongoDB allow for replication across multiple servers, with one server acting as the primary and able to accept writes, and other secondary servers replicating the primary.
- If the primary fails, the replica set will automatically elect a new primary from the secondary servers and continue operating without interruption.
- The replica set configuration specifies the members, their roles, and settings like heartbeat frequency to monitor member health and elect a primary if needed.
Dokumen tersebut membahas tentang limit fungsi aljabar dan trigonometri. Secara singkat, limit fungsi menunjukkan nilai yang didekati oleh suatu fungsi ketika variabelnya mendekati suatu nilai tertentu. Dokumen tersebut menjelaskan langkah-langkah pengerjaan limit fungsi dan beberapa teorema yang terkait dengan limit fungsi aljabar seperti operasi hitung limit dan limit bentuk tak tentu.
Dokumen tersebut memberikan penjelasan tentang pengertian limit fungsi, teorema-teorema limit, dan contoh soal latihan limit fungsi. Limit fungsi didefinisikan sebagai harga batas suatu fungsi ketika variabelnya mendekati nilai tertentu. Ada beberapa teorema limit yang dijelaskan seperti operasi penjumlahan dan perkalian limit, serta contoh soal latihan untuk memahami konsep limit fungsi.
1. Limit fungsi mendekati nilai tertentu (A) ketika variabel mendekati nilai tertentu (a).
2. Langkah mengerjakan limit fungsi antara lain subtitusi langsung, faktorisasi, membagi dengan variabel pangkat tertinggi.
3. Terdapat beberapa teorema limit seperti penjumlahan dan perkalian limit.
1) Dokumen tersebut membahas tentang materi Matematika II khususnya tentang limit fungsi. Dijelaskan definisi dan cara menyelesaikan limit fungsi aljabar dan ketika mendekati tak hingga serta beberapa teorema terkait limit fungsi.
2) Terdapat contoh soal dan penyelesaian mengenai penentuan nilai limit fungsi trigonometri, aljabar, dan ketika mendekati tak hingga.
3) Dibahas pula definisi kontinuitas dan diskontinuit
Dokumen tersebut membahas tentang limit fungsi, termasuk definisi limit fungsi, langkah-langkah pengerjaan limit fungsi, teorema-teorema limit fungsi, dan contoh soal latihan tentang penentuan nilai limit fungsi.
Dokumen tersebut membahas tentang limit fungsi, termasuk definisi limit fungsi, langkah-langkah pengerjaan limit fungsi, teorema-teorema limit fungsi, dan contoh soal latihan tentang penentuan nilai limit fungsi.
Dalam Modul ini, kita mempelajari tentang :
Arti Limit Fungsi di satu titik melalui perhitungan nilai-nilai di sekitar titik tersebut.
Arti Limit Fungsi di tak berhingga melalui grafik dan perhitungan.
Limit Fungsi Aljabar dan Trigonometri di satu titik
Sifat-sifat yang digunakan dalam perhitungan Limit
Arti bentuk tak tentu dari Limit Fungsi.
Menggunakan Sifat-sifat Limit untuk menyelesaikan Limit Fungsi Aljabar dan Trigonometri
Fungsi dan limit memiliki tiga kalimat utama:
1. Fungsi adalah aturan korespondensi yang menghubungkan setiap objek dalam daerah asal dengan nilai tunggal dalam daerah hasil.
2. Limit menggambarkan perilaku fungsi ketika peubah bebas mendekati nilai tertentu.
3. Ada beberapa jenis limit seperti limit ketika x mendekati a, tak hingga, atau nol.
Dokumen tersebut membahas tentang limit fungsi. Secara singkat, limit fungsi menunjukkan nilai yang diraih oleh suatu fungsi ketika variabel mendekati suatu nilai tertentu. Dokumen tersebut juga menjelaskan langkah-langkah mengerjakan soal limit fungsi serta beberapa teorema yang terkait dengan limit fungsi.
Dokumen tersebut membahas tentang limit fungsi. Secara singkat, limit fungsi menunjukkan nilai yang diraih oleh suatu fungsi ketika variabel mendekati suatu nilai tertentu. Dokumen tersebut juga menjelaskan langkah-langkah mengerjakan soal limit fungsi serta beberapa teorema yang terkait dengan limit fungsi.
Stream Processing with Flink and Stream Sharingconfluent
Stream Processing with Apache Flink in the Cloud and Stream Sharing
Kai Waehner discusses stream processing use cases with Apache Flink and introduces Confluent's plans to offer a serverless Apache Flink service in Confluent Cloud. He also discusses Confluent Stream Sharing, which allows for easy and secure sharing of Kafka topics across organizations in real-time.
This document provides information about MongoDB replication and sharding. It discusses what replication is, how to set up replication on Windows including starting primary and secondary servers and verifying replication. It also discusses best practices for replication including always using replica sets, using replica sets to offload reads from primary, and using an odd number of replicas. The document also discusses how to set up MongoDB replication on Linux in a step-by-step process and how to check the replication status. It provides commands for adding and removing MongoDB instances from a replica set and making a primary secondary. Finally, it discusses what sharding is in MongoDB, the concept of sharding keys, and provides a high-level overview of implementing sharding in MongoDB including using
The slides of my talk about Vertical vs Horizontal Software Architecture at Equal Expert's Expert Talks Pune on 2019-09-14. It describes why vertical architectures are better than horizontal architectures for the perspective of change and architecture principles, and how this should reflect in the structure of source code.
- Replica sets in MongoDB allow for replication across multiple servers, with one server acting as the primary and able to accept writes, and other secondary servers replicating the primary.
- If the primary fails, the replica set will automatically elect a new primary from the secondary servers and continue operating without interruption.
- The replica set configuration specifies the members, their roles, and settings like heartbeat frequency to monitor member health and elect a primary if needed.
Dokumen tersebut membahas tentang limit fungsi aljabar dan trigonometri. Secara singkat, limit fungsi menunjukkan nilai yang didekati oleh suatu fungsi ketika variabelnya mendekati suatu nilai tertentu. Dokumen tersebut menjelaskan langkah-langkah pengerjaan limit fungsi dan beberapa teorema yang terkait dengan limit fungsi aljabar seperti operasi hitung limit dan limit bentuk tak tentu.
Dokumen tersebut memberikan penjelasan tentang pengertian limit fungsi, teorema-teorema limit, dan contoh soal latihan limit fungsi. Limit fungsi didefinisikan sebagai harga batas suatu fungsi ketika variabelnya mendekati nilai tertentu. Ada beberapa teorema limit yang dijelaskan seperti operasi penjumlahan dan perkalian limit, serta contoh soal latihan untuk memahami konsep limit fungsi.
1. Limit fungsi mendekati nilai tertentu (A) ketika variabel mendekati nilai tertentu (a).
2. Langkah mengerjakan limit fungsi antara lain subtitusi langsung, faktorisasi, membagi dengan variabel pangkat tertinggi.
3. Terdapat beberapa teorema limit seperti penjumlahan dan perkalian limit.
1) Dokumen tersebut membahas tentang materi Matematika II khususnya tentang limit fungsi. Dijelaskan definisi dan cara menyelesaikan limit fungsi aljabar dan ketika mendekati tak hingga serta beberapa teorema terkait limit fungsi.
2) Terdapat contoh soal dan penyelesaian mengenai penentuan nilai limit fungsi trigonometri, aljabar, dan ketika mendekati tak hingga.
3) Dibahas pula definisi kontinuitas dan diskontinuit
Dokumen tersebut membahas tentang limit fungsi, termasuk definisi limit fungsi, langkah-langkah pengerjaan limit fungsi, teorema-teorema limit fungsi, dan contoh soal latihan tentang penentuan nilai limit fungsi.
Dokumen tersebut membahas tentang limit fungsi, termasuk definisi limit fungsi, langkah-langkah pengerjaan limit fungsi, teorema-teorema limit fungsi, dan contoh soal latihan tentang penentuan nilai limit fungsi.
Dalam Modul ini, kita mempelajari tentang :
Arti Limit Fungsi di satu titik melalui perhitungan nilai-nilai di sekitar titik tersebut.
Arti Limit Fungsi di tak berhingga melalui grafik dan perhitungan.
Limit Fungsi Aljabar dan Trigonometri di satu titik
Sifat-sifat yang digunakan dalam perhitungan Limit
Arti bentuk tak tentu dari Limit Fungsi.
Menggunakan Sifat-sifat Limit untuk menyelesaikan Limit Fungsi Aljabar dan Trigonometri
Fungsi dan limit memiliki tiga kalimat utama:
1. Fungsi adalah aturan korespondensi yang menghubungkan setiap objek dalam daerah asal dengan nilai tunggal dalam daerah hasil.
2. Limit menggambarkan perilaku fungsi ketika peubah bebas mendekati nilai tertentu.
3. Ada beberapa jenis limit seperti limit ketika x mendekati a, tak hingga, atau nol.
Dokumen tersebut membahas tentang limit fungsi. Secara singkat, limit fungsi menunjukkan nilai yang diraih oleh suatu fungsi ketika variabel mendekati suatu nilai tertentu. Dokumen tersebut juga menjelaskan langkah-langkah mengerjakan soal limit fungsi serta beberapa teorema yang terkait dengan limit fungsi.
Dokumen tersebut membahas tentang limit fungsi. Secara singkat, limit fungsi menunjukkan nilai yang diraih oleh suatu fungsi ketika variabel mendekati suatu nilai tertentu. Dokumen tersebut juga menjelaskan langkah-langkah mengerjakan soal limit fungsi serta beberapa teorema yang terkait dengan limit fungsi.
Limit merupakan pendekatan nilai fungsi ketika variabel mendekati suatu nilai tertentu. Terdapat tiga bentuk hasil limit yaitu bentuk tentu, tak tentu, dan tidak terdefinisi. Beberapa teorema limit dapat digunakan untuk menyelesaikan soal-soal limit, seperti penggunaan subtitusi langsung, pemfaktoran, dan membagi dengan variabel pangkat tertinggi ketika variabel mendekati tak hingga.
Fungsi turunan dan integral memiliki peran penting dalam menganalisis grafik fungsi dan menentukan sifat-sifatnya seperti asimtot, kemonotonan, ekstrim, kecekungan, dan titik belok. Uji turunan pertama dan kedua digunakan untuk menentukan sifat-sifat tersebut.
Limit fungsi memberikan pengertian tentang nilai fungsi ketika variabel mendekati nilai tertentu atau tak berhingga. Ada beberapa cara menyelesaikan perhitungan limit fungsi aljabar yaitu dengan substitusi langsung, faktorisasi, dan mengalikan dengan bilangan sekawan. Jika variabel menuju tak berhingga, limit diselesaikan dengan membagi pembilang dan penyabut dengan pangkat tertinggi variabel.
Dokumen tersebut membahas tentang evaluasi pembelajaran, yang mencakup definisi evaluasi, ruang lingkup evaluasi dalam perspektif hasil pembelajaran dan sistem pembelajaran, serta prinsip-prinsip dan teknik evaluasi pembelajaran seperti pengukuran, penilaian, tes, dan non tes.
Teks tersebut membahas tentang bangun ruang kubus dan balok, termasuk menjelaskan sifat-sifat, cara menggambar, dan menghitung luas permukaan kubus dan balok. Imah dan teman-temannya diminta membuat dus kotak nasi dan membahas langkah-langkah membuat jaring-jaring kubus dan balok.
Dokumen tersebut membahas tentang denah dan skala. Denah digunakan untuk menggambarkan letak ruangan pada suatu bangunan sebelum dibangun, sedangkan skala digunakan untuk menyesuaikan ukuran gambar dengan ukuran sebenarnya. Denah dan skala penting untuk merancang dan membangun suatu bangunan.
Bab ini membahas tentang kecepatan dan debit. Kecepatan didefinisikan sebagai jarak yang ditempuh dalam satuan waktu tertentu. Debit adalah volume zat cair yang mengalir dalam satuan waktu. Contohnya debit air yang mengalir dari keran air. Bab ini juga menjelaskan tentang satuan-satuan yang terkait dengan kecepatan dan debit serta cara menghitung nilai kecepatan dan debit.
Ringkasan dokumen tersebut adalah:
1. Dokumen tersebut membahas tentang kurikulum 2013 dan implementasinya serta silabus dan RPP tahun 2019.
2. Terdapat biodata penulis dokumen beserta arah pengembangan SMK menurut Kemendikbud.
3. Juga terdapat penjelasan tentang komposisi kompetensi pada tiap jenjang pendidikan dan bagaimana menurunkan indikator pencapaian kompetensi.
Dokumen tersebut membahas tentang dua jenis pertumbuhan makhluk hidup, yaitu yang sudah mirip induk sejak lahir dan yang mengalami perubahan sampai mirip induk. Kupu-kupu dan katak dijelaskan sebagai contoh hewan yang mengalami metamorfosis, yaitu perubahan dari telur menjadi ulat kemudian kepompong sebelum menjadi dewasa. Proses serupa juga dialami ladybug dan kebanyakan serangga.
Darah berfungsi sebagai alat transportasi untuk mengangkut oksigen, zat makanan, dan hormon ke seluruh tubuh, serta membawa bahan sisa untuk dibuang melalui ginjal, kulit dan paru-paru. Darah terdiri atas plasma, sel darah merah, sel darah putih, dan keping darah, dimana masing-masing memiliki peran penting untuk menunjang kesehatan tubuh. Golongan darah A dapat mendonorkan darah kepada semua golong
This document appears to be a scanned receipt from a grocery store listing various food and household items purchased totaling $123.45. The receipt details the items, quantities, and individual prices purchased on a particular date at a specific store location. In summary, this is a grocery receipt documenting a shopping trip and the total cost of the items purchased.
2. Limit fungsi aljabar
LIMIT FUNGSI:
Mendekati hampir, sedikit lagi,
atau
harga batas .
Untuk lebih jelas perhatikan
contoh berikut :
Perhatikan fungsi berikut f(x) =
2x + 1, dengan x € R . Kita akan
menentukan f(x) dengan x ber
Hal.: 2 LIMIT FUNGSI Adaptif
3. Limit fungsi aljabar
Limit fungsi:
Suatu limit f(x) dikatakan
mendekati a
{f(x) a} sebagai suatu limit.
Bila x mendekati a {x a}
Dinotasikan
Lim f(x) = a
x a
Hal.: 3 LIMIT FUNGSI Adaptif
4. Limit fungsi aljabar
Langkat-langkah mengerjakan limit fungsi
(supaya bentuk tak tentu dapat dihindari)
adalah ….
1. Subtitusi langsung.
2. Faktorisasi.
3. Mengalikan dengan bilangan sekawan.
4. Membagi dengan variabel pangkat
tertinggi.
Hal.: 4 LIMIT FUNGSI Adaptif
5. LIMIT FUNGSI ALJABAR
Berapa teorema limit:
Bila Lim f(x) = A dan Lim g(x) = B
x a x a
Maka
1. Lim [k.f(x)] = k Lim f(x)
x a x a
= k. A
2. Lim [f(x)+g(x)] = Lim f(x) + Lim g(x)
x a x a x a
=A+B
Hal.: 5 LIMIT FUNGSI Adaptif
6. Limit fungsi aljabar
3. Lim {f(x) x g(x)}
x a x a
= Lim f(x) x Lim g(x)
x a x a
=AxB
f ( x ) Lim
f ( x)
A
4. Lim g ( x) = x→a
=
x→a Lim g ( x) B
x→a
Hal.: 6 LIMIT FUNGSI Adaptif
7. Limit fungsi aljabar
5. n
Lim [ f ( x)] = Lim f ( x) = A
n n
x→ a x→ a
6.
n n n
Lim f ( x) = Lim f ( x) =
x→ a x→ a
A
Hal.: 7 LIMIT FUNGSI Adaptif
8. Limit fungsi aljabar
Soal latihan:
1. Nilai dari Lim 3x adalah….
x 2
a. 1
b. 2
c. 3
d. 4
e. 6
Hal.: 8 LIMIT FUNGSI Adaptif
9. Limit fungsi aljabar
Pembahasan 1:
Lim 3x = 3(2)
x 2
=6
Pembahasan 2:
Lim 3x = 3 Lim X
x 2 x 2
= 3(2) = 6
Hal.: 9 LIMIT FUNGSI Adaptif
13. Limit fungsi aljabar
3. Nilai dari Lim [6x-2x] adalah….
x 3
a. -6
b. 8
c. 12
d. 14
e. 16
Hal.: 13 LIMIT FUNGSI Adaptif
14. Limit fungsi aljabar
Pembahasan 1:
Lim [6x-2x] = Lim 4x = 4(3) = 12
X 3 x 3
Pembahasan 2:
Lim [6x-2x] = Lim 6x – Lim 2x
X 3 x 3 x 3
= 6(3) – 2(3)
= 18 – 6 = 12
Hal.: 14 LIMIT FUNGSI Adaptif
15. LIMIT FUNGSI ALJABAR
0
Limit fungsi bentuk
0
Jika f(x) = (x-a).h(x)
g(x) = (x-a).k(x)
Maka:
f ( x) ( x − a ).h( x)
Lim
x →a g ( x)
= Lim
x →a ( x − a ).k ( x )
h( x ) h( a )
=Lim k ( x) = k (a)
x →a
Hal.: 15 LIMIT FUNGSI Adaptif
16. LIMIT FUNGSI ALJABAR
Limit Fungsi Bentuk ~
~
Jika diketahui limit tak hingga (~)
Sebagai berikut:
n −1
ax + bx + ... + c
n
Lim px m + qx m−1 + ... + r = R
x→ ~
Maka:
1. R= 0 jika n<m
2. R= a jika n=m
3. R= ~ jika n>m
Hal.: 16 LIMIT FUNGSI Adaptif
17. LIMIT FUNGSI ALJABAR
Limit Fungsi Bentuk (~ - ~)
Lim [ ]
a.
ax + b − px + q = R
~
x→
1. R= ~ jika a>p
2. R= 0 jika a=p
3. R= -~ jika a<p
Hal.: 17 LIMIT FUNGSI Adaptif
18. LIMIT FUNGSI ALJABAR
Lim [ ]
b.
ax 2 + bx + c − px 2 + qx + r = R
x→~
1. R= ~ jika a>p
b− q
2. R= jika a=p
2 a
3. R= -~ jika a<p
Hal.: 18 LIMIT FUNGSI Adaptif
19. LIMIT FUNGSI ALJABAR
Soal latihan:
4. Nilai dari x 4 − 3x 2 + 4 x
Lim 2 x3 − x 2 − 2 x
x→0
adalah….
1
a. 3 d. −
2
b. 2
c. 1 e. -2
Hal.: 19 LIMIT FUNGSI Adaptif
20. LIMIT FUNGSI ALJABAR
Pembahasan:
x − 3x + 4 x
4 2
Lim 2 x 3 − x 2 − 2 x
x →0
0 − 3.0 + 4.0 0
4 2
= =
2.0 − 0 − 2.0 0
3 2
Jika 0 didistribusikan menghasilkan ~
(bukan solusi) sehingga soal diselesaikan dengan
cara faktorisasi
Hal.: 20 LIMIT FUNGSI Adaptif
21. LIMIT FUNGSI ALJABAR
Maka:
x − x +4 x
4
3 2
Lim
x→0 2 x −x −2 x
3 2
=Lim
[ x x 3 − x +4
3 ]
x→[
0 x 2 x −x −2
2
]
x − x +4
3
3
=Lim
x→0 2 x 2 −x −2
0 −0 +4 4
= = =−2
0 −0 −2 −2
Hal.: 21 LIMIT FUNGSI Adaptif
22. LIMIT FUNGSI ALJABAR
Soal latihan:
4. Nilai dari x 4 −3x 2 + 4 x
Lim 2 x 3 − x 2 − 2 x
x →0
adalah….
1
a. 3 d. −
2
b. 2
c. 1 e. -2
Hal.: 22 LIMIT FUNGSI Adaptif
23. LIMIT FUNGSI ALJABAR
5. Nilai dari x2 −4
Lim x →2 x2 + x −6
adalah….
a.1 2
d.
4 5
b.
5 e. −1
3
c.
5
Hal.: 23 LIMIT FUNGSI Adaptif
24. LIMIT FUNGSI ALJABAR
Pembahasan:
2
x −4
Lim x 2 + x − 6
x →2
( x −2)( x +2)
= Lim
x→ 2 ( x −2)( x +3)
x +2 2 +2 4
=Lim x + 3 = 2 + 3 = 5
x →2
Hal.: 24 LIMIT FUNGSI Adaptif
25. LIMIT FUNGSI ALJABAR
5. Nilai dari x −4
2
Lim
x →2 x + x −6
2
adalah….
a.1 2
d.
4 5
b.
5 e. −1
3
c.
5
Hal.: 25 LIMIT FUNGSI Adaptif
26. LIMIT FUNGSI ALJABAR
6. Nilai dari 4 x + 3x − 6
2
Lim 2 x 2 −8 x −1
x →~
adalah ….
a. -6 d. 16
b. 2 e. 32
c. 10
Hal.: 26 LIMIT FUNGSI Adaptif
27. LIMIT FUNGSI ALJABAR
Pembahasan 1:
4 x + 3x − 6
2
Lim 2 x 2 −8 x −1
x →~
2
4x 3x 6 3 6
2
+ 2 − 2 4+ − 2
= x2 x x = x x
2x 8x 1 8 1
− 2 + 2 2− + 2
x 2
x x x x
Hal.: 27 LIMIT FUNGSI Adaptif
29. LIMIT FUNGSI ALBAJAR
Pembahasan 2:
4 x 2 + 3x − 6
Lim 2 x 2 −8 x −1
x →~
Perhatikan bahwa pangkat diatas sama
dengan pangkat bawah sehingga p = q
(p dibagi q)
p 4
L = = =2
q 2
Hal.: 29 LIMIT FUNGSI Adaptif
30. LIMIT FUNGSI ALJABAR
6. Nilai dari 4 x 2 + 3x − 6
Lim 2 x 2 −8 x −1
x →~
adalah ….
a. -6 d. 16
b. 2 e. 32
c. 10
Hal.: 30 LIMIT FUNGSI Adaptif
31. LIMIT FUNGSI ALJABAR
7. Nilai dari
Lim{ 4 x − 2 x + 6 − 4 x + 2 x − 1}
2 2
x→~
adalah….
a. -3 d. 0
b. -2 e. 1
c. -1
Hal.: 31 LIMIT FUNGSI Adaptif
32. LIMIT FUNGSI ALJABAR
Pembahasan:
b−q −2−2 −4
R= = =
2 a 2 4 2.2
−4
= = −1
4
Hal.: 32 LIMIT FUNGSI Adaptif
33. LIMIT FUNGSI ALJABAR
7. Nilai dari
Lim{ 4 x − 2 x + 6 − 4 x + 2 x − 1}
2 2
x→~
adalah….
a. -3 d. 0
b. -2 e. 1
c. -1
Hal.: 33 LIMIT FUNGSI Adaptif
34. Limit fungsi sljabar
8. Nilai dari (8 x − 2)2
Lim (4 x + 1)2
x→ ~
adalah….
a. -4 d. 4
b. 0 e. 8
c. 2
Hal.: 34 LIMIT FUNGSI Adaptif
35. LIMIT FUNGSI ALJABAR
Pembahasan:
(8 x − 2) 2 64 x 2 − 32 x + 4
Lim (4 x −1)2 = Lim 16 x 2 + 8 x +1
x→~
x→~
64
= =4
16
Hal.: 35 LIMIT FUNGSI Adaptif
36. LIMIT FUNGSI ALJABAR
8. Nilai dari
(8 x − 2) 2
Lim (4 x + 1) 2
x→ ~
adalah….
a. -4 d. 4
b. 0 e. 8
c. 2
Hal.: 36 LIMIT FUNGSI Adaptif
37. LIMIT FUNGSI ALJABAR
9. Nilai dari x −x 2
Lim x 2 + 2 x
x →o
adalah….
a. -~ d. 0
b. -2
1 1
c. − e.
2 2
Hal.: 37 LIMIT FUNGSI Adaptif
38. Limit fungsi aljabar
Pembahasan:
x −x2
x( x − 1)
Lim x 2 + 2 x = Lim x( x + 2)
x →0 x →0
x −1 0 −1 1
= Lim = =−
x→0 x + 2 0+2 2
Hal.: 38 LIMIT FUNGSI Adaptif
39. Limit fungsi aljabar
x2 − x
9. Nilai dari Lim 2
x →o x + 2 x
adalah….
a. -~ d. 0
b. -2
1 1
−
c. e. 2
2
Hal.: 39 LIMIT FUNGSI Adaptif
40. Limit fungsi aljabar
10. Nilai dari 6 x − 4 x + 2 x −1
2 2
Lim 3x 4 − 2 x 3 + 5 x + 2
x →~
adalah….
1
a. − d. 2
2
b. 0 e. 3
1
c. 2
Hal.: 40 LIMIT FUNGSI Adaptif
41. Limit fungsi aljabar
Pembahasan:
6 x − 4 x + 2 x −1
2 2
Lim 3x 4 − 2 x 3 + 5 x + 2
x →~
Perhatikan
Pangkat tertinggi diatas 3
Pangkat tertinggi dibawah 4
Jadi n < m
Nilai R = 0
Hal.: 41 LIMIT FUNGSI Adaptif
42. Limit fungsi aljabar
10. Nilai dari 6x − 4x + 2x −1
2 2
Lim 3x 4 − 2 x3 + 5 x + 2
x→ ~
adalah….
1
a. − d. 2
2
b. 0 e. 3
1
c. 2
Hal.: 42 LIMIT FUNGSI Adaptif
43. Limit fungsi aljabar
11. Nilai dari 2 x + 5 x −12
2
Lim 3x 2 −13x − 4
x →−4
adalah….
5 12
a. d.
13 13
8 14
b. e.
13 13
11
c.
13
Hal.: 43 LIMIT FUNGSI Adaptif
44. Limit fungsi aljabar
Pembahasan:
2 x 2 + 5 x −12
Lim 3x 2 −13x − 4
x →−4
( 2 x − 3)( x + 4)
Lim (3x −1)( x + 4)
x →−4
2 x − 3 2(−4) − 3
Lim 3x −1 = 3(−4) −1
x →−4
−11 11
= =
−13 13
Hal.: 44 LIMIT FUNGSI Adaptif
45. Limit fungsi aljabar
12. Nilai dari 2 x + 4 x −10
2
Lim 4 x 2 + 7
x →~
adalah….
1
a. d. -1
2
b. 0 e. -6
c. 1
−
2
Hal.: 45 LIMIT FUNGSI Adaptif
46. Limit fungsi aljabar
Pembahasan: 2 x 2 + 4 x −10
Lim 4 x 2 + 7
x →~
Pangkat diatas = Pangkat dibawah
Maka
2 1
=
4 2
Hal.: 46 LIMIT FUNGSI Adaptif
47. Limit fungsi aljabar
12. Nilai dari 2 x + 4 x −10
2
Lim 4 x 2 +7
x→~
adalah….
a. 1 d. -1
2
b. 0 e. -6
c. − 1
2
Hal.: 47 LIMIT FUNGSI Adaptif
48. Limit Fungsi Trigonometri
1. Bentuk lim f(x) = f(a)
x→a
Contoh :
Tentukan nilai lim sin 2x.
π
x→
4
Jawab :
π π
Lim sin 2x = sin 2 = sin 2 =1
x→
π 4
4
Hal.: 48 LIMIT FUNGSI Adaptif
49. Limit Fungsi Trigonometri
f ( x)
2. Bentuk lim , dengan f(a) = 0 dan g(a) = 0
x → a g ( x)
Ingat !!!
Contoh : sin 2 x
Tentukan nilai dari : lim sin 2 x = 2 sin x cos x
x→
π
2
cos x
Jawab : cos 2 x = 1 − 2 sin 2 x
sin 2 x 2 sin x cos x π
lim = lim = lim 2 sin x = 2 sin = 2.1 = 2
π π π
x → cos x x→ cos x x→ 2
4 4 4
Hal.: 49 LIMIT FUNGSI Adaptif
50. Limit Fungsi Trigonometri
3. Bentuk lim tan x atau lim
sin x
x→0 x x→0 x
Catatan :
sin x x
1. lim = lim =1
x→0 x x → 0 sin x
tan x x
2. x → 0 x x → 0 tan x = 1
lim = lim
Secara umum
sin ax a tan ax a tan ax a
lim = , lim = , lim =
x → 0 bx b x → 0 bx b x → 0sin ax b
Hal.: 50 LIMIT FUNGSI Adaptif
51. Limit Fungsi Trigonometri
Contoh 1 :
Tentukan nilai limit fungsi trigonometri berikut!
sin 8 x tan 3x 1 − cos 2 x
a. lim b. lim c. lim
x→0 2x x→0 sin 4 x x→0 x2
Jawab :
sin 8 x sin 8 x 8
a. Lim = lim . = 1.4 = 4
x→0 2x x →0 8x 2
tan 3x tan 3 x 4 x 3
b. lim = lim . .
x →0 sin 4 x x →0 3 x sin 4 x 4
3 3
= 1 .1 . =
4 4
Hal.: 51 LIMIT FUNGSI Adaptif
52. Limit Fungsi Trigonometri
1 − cos 2 x 1 − (1 − 2 sin 2 x)
c. lim 2
= lim
x →0 x x→0 x2
2 sin 2 x
= lim
x→0 x2
2
sin x
= lim 2.
x →0
x
2
sin x
= 2. lim
x→0 x
= 2.12 = 2
Hal.: 52 LIMIT FUNGSI Adaptif
53. Limit Fungsi Trigonometri
Contoh 2 :
cos 2 x − 1
Tentukan nilai dari lim
x→π sin x
Jawab :
cos 2 x − 1 (1 − 2 sin 2 x) − 1 − 2 sin 2 x
lim = lim = lim
x→π sin x x→π sin x x→π sin x
= lim(−2 sin x) = −2 sin π = −2.0 = 0
x→π
Hal.: 53 LIMIT FUNGSI Adaptif
