Infinite limits and asymptotes

- [Instructor] What we're going to do in this video is use the online graphing calculator Desmos, and explore the relationship between vertical and horizontal asymptotes, and think about how they relate to what we know about limits. So let's first graph two over x minus one, so let me get that one graphed, and so you can immediately see that something interesting happens at x is equal to one. If you were to just substitute x equals one into this expression, you're going to get two over zero, and whenever you get a non-zero thing, over zero, that's a good sign that you might be dealing with a vertical asymptote. In fact we can draw that vertical asymptote right over here at x equals one. But let's think about how that relates to limits. What if we were to explore the limit as x approaches one of f of x is equal to two over x minus one, and we could think about it from the left and from the right, so if we approach one from the left, let me zoom in a little bit over here, so we can see as we approach from the left when x is equal to zero, the f of x would be equal to negative two, when x is equal to point five, f of x is equal to negative of four, and then it just gets more and more negative the closer we get to one from the left. I could really, so I'm not even that close yet if I get to let's say 0.91, I'm still nine hundredths less than one, I'm at negative 22.222, already. And so the limit as we approach one from the left is unbounded, some people would say it goes to negative infinity, but it's really an undefined limit, it is unbounded in the negative direction. And likewise, as we approach from the right, we get unbounded in the positive infinity direction and technically we would say that that limit does not exist. And this would be the case when we're dealing with a vertical asymptote like we see over here. Now let's compare that to a horizontal asymptote where it turns out that the limit actually can exist. So let me delete these or just erase them for now, and so let's look at this function which is a pretty neat function, I made it up right before this video started but it's kind of cool looking, but let's think about the behavior as x approaches infinity. So as x approaches infinity, it looks like our y value or the value of the expression, if we said y is equal to that expression, it looks like it's getting closer and closer and closer to three. And so we could say that we have a horizontal asymptote at y is equal to three, and we could also and there's a more rigorous way of defining it, say that our limit as x approaches infinity is equal of the expression or of the function, is equal to three. Notice my mouse is covering it a little bit as we get larger and larger, we're getting closer and closer to three, in fact we're getting so close now, well here you can see we're getting closer and closer and closer to three. And you could also think about what happens as x approaches negative infinity and here you're getting closer and closer and closer to three from below. Now one thing that's interesting about horizontal asymptotes is you might see that the function actually can cross a horizontal asymptote. It's crossing this horizontal asymptote in this area in between and even as we approach infinity or negative infinity, you can oscillate around that horizontal asymptote. Let me set this up, let me multiply this times sine of x. And so there you have it, we are now oscillating around the horizontal asymptote, and once again this limit can exist even though we keep crossing the horizontal asymptote, we're getting closer and closer and closer to it the larger x gets. And that's actually the key difference between a horizontal and a vertical asymptote. Vertical asymptotes if you're dealing with a function, you're not going to cross it, while with a horizontal asymptote, you could, and you are just getting closer and closer and closer to it as x goes to positive infinity or as x goes to negative infinity.