The definition of a limit continues on in our learning limits and how they are applied. Specifically we are going to look at some proofs in this section. With a precise definition we can now prove many limit properties. Definition of a Limit The \(\delta\) value will always depend on the \(\epsilon\) value. That is important to remember in order to understand what is happening in theselimits. Once you find a value that works in your problem then smaller values of that \(\delta\) also work. Problem 1. For the given function f(x) and values L,c, and \(\epsilon\) > 0 find the largest open interval about c on which the inequality |f(x)-L| < \(\epsilon\) holds. Then determine the largest value for \(\delta\) >0 such that 0<|x-c|<\(\delta\)\(\rightarrow\) |f(x)-L|<\(\epsilon\). f(x) = \(\frac{1}{x}\), L= 0.125, c = 8, \(\epsilon\) = 0.015 Solve |f(x) - L| < \(\epsilon\) to find the largest interval containing c on which the inequality holds. Write the inequality without absolute value. Substitute \(\frac{1}{x}\) for f(x), 0.1215 for L, and 0.015 for \(\epsilon\). |f(x)-L| < \(\epsilon\) |\(\frac{1}{x}\) - 0.125| < 0.015 -0.015 < \(\frac{1}{x}\) - 0.125 < 0.015 Simplify by adding 0.125 to all expressions. -0.015 < \(\frac{1}{x}\) - 0.125 < 0.015 0.11 < \(\frac{1}{x}\) < 0.14 Simplify further by taking the reciprocals of all terms. Be sure to reverse the inequalities. Make sure you check that the rounded endpoints still allow the inequality |f(x)-L| < \(\epsilon\) to hold true. 9.0909 > x > 7.1429 Since the interval 7.1429 < x < 9.0909 is not centered on c=8, \(\delta\) is the distance from 8 to closer endpoint of the interval. The value of \(\delta\) is then 0.8571.   Problem 2. For the given function \(f(x)\), the point \(c\), and a positive number \(\epsilon\), find \(L=lim f(x)\). Then find a number \(\delta\) > 0 such that for all x, 0<|x-c|<\(\delta\)\(\rightarrow\)|\(f(x)-L\)| < \(\epsilon\). \(f(x)=8-3x, c=4, \epsilon=0.03\) Notice that \(f(x)\) is a linear function. Since a linear function is a simple polynomial function and it is defined for all \(x\), the limit \(L\) = lim is the value of \(f(x)\) at \(c\). Evaluate the function at \(c=4\). \(L=8-3(4)=-4\) To find \(\delta\), begin by solving the inequality \(|f(x)-L|<\epsilon\) to find an open interval \((a,b)\) containing \(c\) on which the inequality holds for all \(x eq c\). Remove the absolute value sign and rewrite the inequality as a compound inequality. \(|(8-3x)-(-4)| < 0.03\) \(-0.03 < (8-3x) - (-4) < 0.03\) Simplify the center expression and isolate the x-term. \(-0.03 < -3x+12 < 0.03\) \(-12.03 < -3x < -11.97\) Then isolate \(x\) in the center. Notice that the direction of the inequalities has been changed. \(4.01 > x > 3.99\) Therefore, for \(x\) in the interval \((3.99,4.01)\), the inequality \((8-3x)-(-4)|<0.03\) holds. Now choose a positive value for \(\delta\) that places the open interval \((c-\delta,c+\delta)\) centered on \(c\) inside the         interval \((3.99,4.01)\). The distance to the first endpoint is \(4-3.99=0.01\). The distance to the second endpoint is \(4.01 -4=0.01\). The largest possible value for \(\delta\) is \( 0.01\). Therefore, for all \(x\) satisfying \(0<|x-4|<0.01\), the inequality \(|(8-3x)-(-4)|<0.03\) holds.   Problem 3. Give an \(\epsilon - \delta\) proof of the limit fact. \(lim_{x \to 0}(5x+1)=1\) We begin by giving the precise meaning of a limit. To say that \(lim f(x)=L\) means that for each \(\epsilon>0\) there is a corresponding \(\delta>0\) such that \(|f(x)-L|<\epsilon\), provided that \(0<|x-c|<\delta\). To establish the proof, we first perform a preliminary analysis to find the appropriate choice of \(\delta\). Let \(\epsilon\) be any positive number. We must produce a \(\delta > 0\) such that \( 0<|x-0|<\delta \rightarrow |(5x+1) -1 < \epsilon\). By simplifying the inequality on the right, we will determine the value of \(\delta\) needed for the inequality on the left. \(|(5x+1)-1| < \epsilon \Leftrightarrow |5x| < \epsilon \) \(\Leftrightarrow |5| |x| < \epsilon\) \(\Leftrightarrow 5|x| < \epsilon\) \(\Leftrightarrow |x| < \frac{\epsilon}{5}\). Now we see that we should choose \(\delta=\frac{\epsilon}{5}\). We can now construct the formal proof of the statement \(\lim_{x \to 0}(5x+1)=1\). Let \(\epsilon > 0\) be given. Choose \(\delta=\frac{\epsilon}{5}\). Then \(0<|x-0|<\delta\) implies the following chain of equalities and an inequality. \(|(5x+1)-1| = |5x|\)   Simplify inside the absolute value bars. \(           = |5| |x|\)    Rewrite as product of absolute values. \(           = 5 |x| \)     Simplify. \(           =5 |x-0|\)    Rewrite expression inside absolute value bars. \(           < 5\delta\)  Apply the condition \(0<|x-0|<\delta\). \(           = \epsilon \)  Substitute \(\delta=\frac{\epsilon}{5}\). The result of the above chain of equalities and an inequality is that \(|(5x+1)-1|< \epsilon\). Therefore, we have proven that \(\lim_{x \to 0}(5x+1)=1\). Conclusion The definition of a limit has some nice proofs that can establish. If you go over these problems slowly the ideas behind limits will begin to emerge. While this section is short it is essential to understanding limits and beginningcalculusin general.         ...read more

This is my Step by Step Excel 2016 Tutorial. Excel 2016 is new and it can be confusing. I was told this recently and I totally agree with that statement. It looks and feels different than the previous versions. If you have always been a little afraid or unsure of how to use Microsoft Excel well, I wrote this step by step Excel 2016 tutorial just for you! After reading this and going through my examples I promise you will have a much better grasp on how to use Excel spreadsheets. Read the rest at :http://aindien.com/step-by-step-excel-2016-tutorial/ ...read more

Network Basics: Easy Addressing and Routing is all about the TCP/IP protocols and what they do. Our present day internet is totally dependent on this system because because of how it developed and without a doubt it can be confusing. Think about it: The reason TCP/IP was used was because it was rock solid. However, it seems like there are a million details to it and I am sure you would agree with me. The addressing and routing that moves all of our internet traffic between Facebook, Twitter, and all of our other favorite websites uses TCP/IP primarily so there has to be a reason for it right? There really is so let me explain to it to you. Easy Addressing and Routing IP addressing is the system that dictates how every device communicates with each other over a private network or the internet as a whole. Every device must have a unique name and be identified thusly so all our computers who to talk to. If you find this article interesting then check out the full page at: http://aindien.com/network-basics-easy-addressing-and-routing/ ...read more

Everyone knows about Google and nearly everyone knows about Google Apps. It is the second most used software for Office suites. One of the most used within their system are the documents or Docs for short. It is their cool word processing app that you can use online from any computer. In this guide I show how Google Docs work and how to make the most of this great software. Browser Based The whole Google apps system is browser based and this means mainstream web browsers too. We are talking about Internet Explorer, Edge, Firefox, and Chrome of course. You should stick to these main browsers so that you will not have any issues with certain features working. Google Docs Features What Google Docs lacks in general option in how to do things, it totally makes up in subtle but well thought out uses for their software. What I mean by that is there may not be five ways to do everything so every person can do things according to their preference. Instead, though, there are much less ways to do a particular task. To offset this Docs has several really cool features that are unique to it. However, I want to point out that even though it has less features than some types ofproductivity softwareit is always getting more features and in constant development. How Google Docs Work The first thing you need to do is log in to your Google account. This is on Google's homepage and several other places too. Once you are logged in then go to yourDrive areathat will hold all of your newly created files. You can also get toDocsby going directly to it. Just use that link I provided. [caption id="attachment_487" align="alignnone" width="1024"]how google docs work[/caption] You will notice the big blue button at the top left because that is where you go to create new files. Just do this: Click blue button Select Google Docs Click blank document Your new document will open and you will then see the familiar options for most word processors. Now when you create a new document you can just create a blank document like we first did or you can create a document using a template. This means it will already be styled and formatted a little and is helpful if you need a certain type of document such as a report or an essay. There are many other options also so just choose what fits your need the best. Using a template is also nice because it saves you time in creating your document from scratch. Since there is also not a save button using Docs by Google, it gets saved constantly by the system which makes that a very nice feature. There is no more losing your work because your computer crashes and you lose your work. Formatting Documents Formatting is easily done using Google Docs just like with any other word processing suite. There is a nice toolbar which has every option that you would ever need and furthermore you have the menu system up to that will do the same thing. [caption id="attachment_489" align="alignnone" width="1007"]how google docs work[/caption] If you prefer the menu system then just use the [Format] option because it will do the same thing as the icons down below. Speaking of the icons, you can see they are all grouped together in usual ways. Though they are in a slightly different order than other suites, it is easy to see their function. From the icon bar you can print, change your font, change font size, make words bold, center the text, and even make bullet points quite easily. There is a lot more that you can do but those are the usual basics which most people will need. Finding Extra Fonts Google Docs has a lot of different font options that are quick to see and use. However, if you still do not see something that you like then you can always click [More font...] at the bottom of the font list. There are a lot more here and what is really cool is that you can sort them by their popularity. [caption id="attachment_490" align="alignnone" width="600"]how google docs work[/caption] It is nice to see what is currently popular among others. The list to the right is also very useful as it is your list of fonts that you like and have used. This makes keeping track of them very easy which is a useful feature. Display Styles There are some different styles to apply to your writing. While there are not as many as some suites they are still nice. You also have the option of saving your style and making it a template which is a useful feature. If you do a lot of writing I would recommend that you do that because it will save you time. Lists We have the usual list options here as well and that is always nice. Google Docs has several varieties of bulleted and numbered lists to choose from. They are easy to apply and create because you can just hit the icon for each. If you click the tiny little arrow next to either of them you will see six different options for both bulleted and numbered lists. They are all nice to choose from and you can click on one of the styles to select them. Inserting Content The insertion of most content into your document is really quite easy. Just look at your menu and choose the [Insert] tab. You have many options to choose from including links, images, equations, and tables. This makes your document look nice and help the user that is reading it to understand easier what your message is about. If you insert images keep in mind that generally around 2 mb is your limit for an image so if some are larger than that then you will need to reduce their size some. If you need to do links and are not experienced at this their is a useful trick to it. You should go to the site that you want to link to and high the address at the top in the address bar. Once it is highlighted you just copy it to your clipboard and in your document you paste it in to the link box. Document Management Google makes managing your documents easy like most word processing suites. There is the usual list of opening, renaming, and moving options here as well. Nothing too exciting about that anyway. There are a couple features that stand out and I want to mention them. The first is the [Share] option which makes it easy to send a friend the copy of your document. You can just enter your friend's email address and that is it. It is a good feature if you need to collaborate with someone like an editor to fix your grammar mistakes. I know nothing about that, really I don't. The other cool feature I wanted to show you was their [Revision] option. This makes it nice to see how your work has changed and to correct things if you have suddenly gone off on a huge tangent in your new fiction novel. It lists my two revisions there and I would just select them to see the differences. They open right up depending on which one you select. The Toolbox The tool section of just about any app these days has the most useful items for me. It is also where I go to for every document that I write because there is always something I need from there before I finalize any document that I write. It has a decent spellchecker you should always use. While any automated spellchecker can help you just remember they might not catch every mistake you make. The word count feature is always nice because it can tell you where you are at quickly. The [Voice typing] feature is also quite handy as it will dictate for you. It is not perfect but it can save you time initially as you fill out your document. Then you just go back and check for errors but you will have still saved time. I do encourage you to try this option out. Web Publishing When you finish your document there is another option for sharing it and that is publishing it somewhere like a website or social media. This is also under the [File] tab on the menu and you just select the [Publish to the Web] option. You can use the link tab there to share with others on any type of social media. The tab is there so you can insert this link to your document on a website which is pretty cool. Use Operators In Searching This is your main Google Docs page and its also called your home page. If you have dozens, hundreds, or even thousands of documents stored here don't go scrolling down and looking one by one. Use the search bar at the top and use Google's ownsearch operatorsto help you out. The basics in that link are at the top but there are more advanced options down below. Downloading Your Documents Downloading any document is an easy task to do and it makes your document available offline. This especially useful if you are going to be without internet for a while and need to keep working. From the menu select File Then choose Download as... Now choose what format you would like It will then go to your default download folder wherever that is for you. Uploading Documents Once you have worked on a document offline you will need to get it back into the Google Docs system. This is easy to do as well and I will show you how. Select the little folder in the picture that is at the far right. Once you have done that you will then select the [Upload] tab on the far right. Once you click upload you can either drag a document in there or browse to some folder and select it. Easy! Use Templates Templates are one of those features that everyone should use. If you create documents then you should be using templates because it is that simple. I mentioned them earlier but I wanted to go in a bit more detail here. They are on your Docs homepage and labeled nicely so it is easy to see what they are. This is the first page of the template section and you can see there are a lot of options. Resumes and letters are probably the most often created form of document and therefore that is why they are at the top here. The next time you need to create one of these types then just browse through a few of these to find what you like. Table Of Contents If you have a very long document that includes many sections then it would be very handy to use a table of contents. You can do this by adding it as an add-on. Let me show you how. With your document open, select Add-ons from the menu. Then you will choose Get Add-on. In the search box, type out table of contents and hit enter. Researching Content This is my favorite feature of Google Docs because I do a lot of research. It is now called the [Explore] option and you get to it from the [Tools] tab in your main menu. It instantly analyzes your document and suggests topics for you to explore. You can also just use the search bar too which is very nice. When you use this function you will notice that you do not ever have to leave your document and that means no more switching between pages constantly. Make Footnotes This is very easy to do in Google Docs and if you are writing the kind of document that requires them then you are in luck. Select [Insert] from the main menu. Choose the [Footnote] option. Then just type your footnote out and it goes where it should at the bottom. Image Editing You can even do images when you are using Google Docs. After you insert one the options to crop and resize will show up and you can make your images look exactly how you like. Just remember not to have them too big or they will not be able to upload. Using Maps Mapsare similar to images in that you insert them. Most people, however, do not think of inserting a map into their document but its an entirely useful thing to do. You do it just like inserting a picture into your document except that you search by maps or locations. This will bring up maps into your search results and then you just insert that. It is very useful to do that if your document is talking about a location. Conclusion The nice thing about how google docs work is that everything is stored in the cloud and that makes it available everywhere to you. For all of you that work on the road or just the train this isa critical factor. While it does not have all the options of theMicrosoft Officesuite it is quite adequate for most people. This makes it easier to use and therefore less distracting when your in the writing zone and putting out good content. ...read more

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Problems Switching From Word 2010 to Word 2013 http://aindien.com/problems-switching-from-word-2010-to-wo…/ If you find this helpful please subscribe to my newsletter and share this post with your friends. ...read more

I like using Office and Windows products in general. They have strengths that are hard to compete with. Here are some of the reasons. http://aindien.com/taking-advantage-of-outlook-2013/ If you find this helpful please subscribe to my newsletter and share this post with your friends. ...read more

Windows tiles was a big step for Microsoft when they introduced Windows 8. Since a lot of people still use this I wanted to talk a bit about them today. http://aindien.com/working-with-tiles-in-windows-8-1/ Please subscribe to my email list on my website and I would appreciate it greatly if friends and family shared this with your friends too. ...read more

For those of you who use and love Linux, working with files is a very important activity whether pictures or writing a paper. Learn how to change permissions and control what can be done with them. http://aindien.com/managing-linux-files-and-their-permissi…/ Please subscribe to my email list on my website and I would appreciate it greatly if friends and family shared this with your friends too. ...read more

The Reasons Why Networking Your Printer Is A Good Idea http://aindien.com/the-reasons-why-networking-your-printer…/ I would greatly appreciate it if friends and family share this article. I have share buttons on my website to make it easy for you below the title of every post. ...read more

For those who want to customize your Outlook 2013 application I go over what options you have. http://aindien.com/customizing-outlook-2013/ ...read more

Here is a Windows tip for customizing your task bar. I love customizing and finding interesting ways to use software. http://aindien.com/adjusting-the-task-bar-properties-in-windows-8-1/g The Task Bar Properties In Windows 8.1 ...read more

Networking is the foundation of the internet and all that we do on it. Here is a little about why it works. http://aindien.com/understanding-networking-protocols/ ...read more

Most people I know do not know some of the basics of using Google search. In these days where "Googling" is so ubiquitous it is very handy to know some of these tips. I explain one of them here. http://aindien.com/searching-specific-sites-and-categories…/ If you enjoy these tips please subscribe to my newsletter on any page ofwww.aindien.com ...read more

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