 Hello, and welcome to the screencast, which is a sort of part two of finding limits using tables. In this video, we're going to do a very quick introduction to using spreadsheets in the context of limit calculations. The motivation here is that when determining whether a function has a limit, it's often simple to tell if the limit exists, but not so easy to tell what the value of the limit is with any degree of accuracy. Tables of data give us more accuracy than graphs many times, but we will also sometimes need very large tables to get really good accuracy. And since this is the 21st century, we don't fill out long tables by hand using a calculator, but rather we enlist a computer tool to automate that process, and that would be the spreadsheet. Now, a spreadsheet is a computer tool that allows us to make very quick computations on large sets of numerical data. It is an essential piece of software for those in science, engineering, and business, and so we're going to be using spreadsheets quite a lot here in calculus as well. Here I have open Microsoft Excel, which is a standard spreadsheet you can find on almost any computer. If you don't have Microsoft Excel, or if you prefer not to use it, there are other alternatives. For example, Google Docs has an online cloud-based spreadsheet that links with Google Drive, and there is a free open-source spreadsheet called OpenOfficeCalc, which you can download for free at openoffice.org. Even Geogebra has a spreadsheet built into it, but we'll stick to Excel for now since it's an industry standard. So I'm not going to go into depth here about how to use spreadsheets, but rather we're just going to see how to replicate the table calculations that we saw in the handwritten examples that were in the previous screencasts. If you want to know more about Excel, there are many, many places to go and learn about it online, and we'll be doing some of that in class. First, I'm going to set up my table here for x approaching 0 from the left. I am entering in by hand the values x equals negative 1, negative 0.5, negative 0.1, negative 0.01, and I'm going to go a little bit farther and put in negative 0.001, negative 0.0001, and even negative 0.0001. Those are my x values. Notice they are all in column A starting row 1. Now each cell in a spreadsheet has an address that we can use to refer to it. For example, this cell is A2, that is column A row 2. Now let's go up to cell B1 and do the following. First, type in an equal sign, but do not press enter yet. If you make a mistake or if you happen to press enter, just click on the cell to highlight it and then hit the delete key on your keyboard to wipe out what's in the cell. Again, after the equal sign, type parenthesis 2, then the caret key for an exponent. Don't hit enter yet, and then click on the cell A1 or type in A1. Notice that when you click on that cell, Excel will put in A1 into the formula that we're creating. Now continuing, type minus 1, close the parenthesis, then the division symbol, and then click A1 again or type A1 and do not hit enter just yet. Notice what we have here is our formula for f of x, except instead of using a variable x, I have a cell value that I'm substituting in for x. Now we will hit enter and notice that Excel does the calculation that we did by hand in the previous video. But what's really the killer feature of spreadsheets is this. Click on the cell B1 that you just calculated. Notice there's a little square at the bottom right of the cell. Hover your cursor over that square so that the cursor turns into a plus sign. Click and hold on that plus sign, and while you're holding, drag down through the entire column and then let go. Notice that Excel applies the formula that you typed into cell B1 throughout all the cells you just dragged over and intelligently substitutes the new values of x into column A. If you click on cell B4, for example, notice that it's the same formula, but it's using cell A4 for the input. So now we very quickly created a big table of values to help determine our limit. Let's do this again for the values of x approaching zero from the right. Manually enter in the x values you want. Then in the cell next to the top entry, type in the formula for f, making sure to begin with an equal sign. And don't hit enter until you're done and make sure your parentheses match. Hit enter to calculate the value of that cell. Then drag the formula down the column to create the formula for f over all the cells. From these two tables, we get a very accurate read on the value of the limit of f of x as x approaches zero. Both tables have stabilized out to the 10 to the negative fourth place. So we can say confidently that at least a four decimal places of accuracy, the limit of 2 to the x minus 1 over x, exists and is equal to 0.6931. If we wanted more accuracy than just four decimal places worth, we could just simply build a larger table, and this is very easy to do. In fact, the value of this limit is exactly equal to the natural logarithm of 2, which equals this number out to 10 decimal places. But we have a little ways to go before we can figure out why that's the case. Thanks for watching.