 Welcome to the GVSU Calculus Screencasts. In this episode, we talk about limits of sequences. Recall that a sequence is just a list of terms in some specified order, or, more formally, it's a function whose domain is a set of positive integers. In Calculus I, we describe what it meant for a continuous function to have a limited infinity. And now we're going to see how to translate that idea of a limited infinity for a continuous function to a limit of a sequence. So let's go back to Calc 1 and review the idea of a limit of a function, a continuous function at infinity. Let's consider the example of the function f, defined by f of x equals 1 over x. And the graph of 1 over x is that familiar hyperbolic shape. Now pause the video for a moment and consider if f of x has a limit at infinity. If yes, why, and what's the limit? And if no, why not? Resume the video when you're ready. Remember that if we can make the values of f of x as close to some number as we want by choosing x to be large enough, then our function f has a limit at infinity. In this case, we can make all the values 1 over x as close to 0 as we like by choosing x to be sufficiently large. So we say that the limit of f of x as x approaches infinity is 0. Now let's translate this into the sequence world, and let's consider the sequence s sub n, where s sub n is 1 over n for each positive integer n. We can plot the sequence as a function whose points are of the form n s sub n. After all, a sequence is a function from the positive integers to the reals. For example, the point 1 s sub 1, which is a point 1 1, is plotted here, along with the graph of the continuous function 1 over x for comparison purposes. We can plot more points in the sequence to get a sense of the behavior of the sequence, and we'll plot the first five points in the sequence on the next few slides. Here's the first point, 1 1. This is the second point, 2 and a half. Now 3 and a third. And 4 and 1 quarter. And we had 5 and 1 fifth. And you can probably see the process. Here we'll plot the first 25 points in the sequence. And here you can see that the sequence follows the same overall pattern as the function 1 over x, except only hitting the integer inputs. And notice that just as with 1 over x, we can make all the values in our sequence 1 over n as close to 0 as we like, as long as we go far enough out in the sequence. And going far enough out in the sequence means choosing n to be as large as we need. So we can translate the idea of a limit of a continuous function to that of a sequence by saying that our sequence, in this case 1 over n, has a limit of 0 because we can make the values of 1 over n as close to 0 as we like by choosing n to be sufficiently large. We can do this for any sequence. We'll say that a sequence s sub n has a limit of some number l. If we can make all the values of s sub n as close to l as we like by choosing n to be large enough. When this happens, we write limit of s sub n as n goes to infinity is equal to l. And we also say that the sequence s sub n converges to l. So as a concluding example, let's consider the sequence n over n plus 1. Pause the video for a moment and think about this sequence and make a conjecture as to whether this sequence has a limit or not. If so, why and what's the limit? And if not, why not? Here's a plot of points in the sequence n over n plus 1. And this graph seems to indicate that we can make all the values of n over n plus 1 as close to 1 as we like as long as we choose n to be big enough. And so that indicates that the limit of the sequence n over 1 plus n is 1. The main idea to come away with from this screencast is that a sequence s sub n has a limit l if we can make the values of s sub n as close to l as we like by choosing n to be sufficiently large. That concludes our screencast on limits of sequences and we hope to see you again soon.