 Hi, and welcome to the screencast on how to compute limits of functions using tables. We've already discussed the concept of a limit. Specifically, we say that a function f of x has a limit l as x approaches a, if the values of f of x approach l as x approaches a. We've seen how to determine if a function has a limit using a graph or using a graphing tool. Let's look at the same problem from a different point of view using tables of numerical data instead. So for our example, let's look at the function f of x equals 2 to the x power minus 1 divided by x. And let's determine if f has a limit as x approaches 0. As with previous examples, note that f does not exist at x equals 0. Because substituting x equals 0 into my formula would lead to division by 0. But this doesn't tell us that the limit as x approaches 0 fails to exist. It might exist and it might not exist. We need to see what f of x does as x approaches or gets closer and closer to 0 before we can make that determination. First of all, let's have you do a little reviewing. Take your graphing tool, whether it's geogibra or graphing calculator or something else. And I want you to plot a graph of this function, 2 to the x minus 1 divided by x, and make sure that you can see the function's graph around x equals 0. Once you've done this, come back and answer the following multiple choice question about the value of the limit of f as x approaches 0. So pause the video, go make your graph, then choose what you think is the right answer, and then come back and unpause the video to continue. So the correct answer here is B. The limit of f as x approaches 0 does exist and is somewhere between 0 and 1. Here's a graph that should tell us that. Although the graph should technically have a hole in it right here at x equals 0, as x approaches 0, the function is converging on a single output value from both the left and the right, and pretty clearly this value is between 0 and 1. However, what is that value exactly? From this picture it's a little hard to say what the limit actually equals. You might guess around two thirds or three fourths, but without zooming in further on the graph, it's difficult to say exactly what the limit equals. We can only see that it exists. So this example reveals a sort of a downside to determining limits through graphical means. It's sometimes, quite often, easy to tell if a limit exists, but it's not always straightforward to tell what the value of that limit actually is. So let's think about limits, not from a graphical point of view, where the value of the limit might be hard to tell, but from a numerical point of view, where it might be a little bit easier. So we've already determined that this function 2 to the x minus 1 divided by x has a limit as x approaches 0. But the thing is that we don't really know exactly what it is. It's kind of hard to even guess what it is. So what we're going to do here is instead of using graphs, which have some inherent inaccuracies in them because of the way that we see things, we're going to create two tables of data instead that do the same thing that the graph is doing. Namely, I've got two tables here, one for x approaching 0 from the left, the other for x approaching 0 from the right. And we're just going to calculate the values of f of x at these values of x that are approaching 0 and see what the values of f of x are approaching. So this is just a matter of making several repeated calculations for values of x that are approaching but never quite equal to 0. I'm going to do one of these and then we'll jump to the answers here because all the calculations are basically the same. So let's start here on the left, where we have x approaching 0 from the left. And let's just pick a point that's sort of kind of close to 0 but not right on top of it, but not too far away either. Namely, x equals negative 1. Now if you compute f of negative 1, it will be 2 to the minus 1, minus 1 over negative 1. And that turns out to be positive 0.5. Okay, that gives us no information about what the limit as x approaches 0 of this function actually is. We really need to fill out this entire table here, maybe even more entries that are in the table. If I go to x equals negative 0.5, I would do a similar calculation and the result comes out to be 0.58579. Still can't tell exactly what that limit is approaching, it's a little hard to say. So let's go one step farther to negative 0.1. So much closer to 0 now and our result comes out to be 0.66967. If I go to negative 0.1, it's 0.69075. So notice that the outputs here have stabilized at the tenths place. So I believe that the limit as x approaches 0, at least coming from the left is going to be 0.6 something. I'm going to pause the video for a second and go off and calculate all the x values that are approaching 0 from the right, and we'll be right back. Okay, and so here are the values of f of x as x approaches 0 from the right. Now these two values here that are just a hundredth of a unit away from 0, you can begin to see they're converging toward the same value. Something around 0.69. So whereas we already knew that the limit as x approaches 0 of this function exists with a table approach to calculating the limit, we can see a little bit better picture of what the limit actually is going to equal, maybe something around 0.69. Now if we wanted even further accuracy here, we should probably make a really large table that goes to say x equals 0.001 and maybe even many more zeros after that. That begins to get very tedious. And so let's switch over to a great tool for us to use here in calculus and that is the spreadsheet. And since this video is already at six minutes, we're going to make that into a separate screencast, so stay tuned.