Infinite limits and asymptotes | Limits and continuity | AP Calculus AB | Khan Academy

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 ( \frac{2}{x - 1} ). So let me get that one graphed. You can immediately see that something interesting happens at ( x ) is equal to 1. If you were to just substitute ( x ) at 1 into this expression, you're going to get ( \frac{2}{0} ). 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 = 1 ).

But let's think about how that relates to limits. What if we were to explore the limit as ( x ) approaches one of ( f(x) ) is equal to ( \frac{2}{x - 1} )? We can 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 1, ( f(x) ) would equal to -2. When ( x ) is equal to 0.5, ( f(x) ) is equal to 4, 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 0.09 less than one. I'm at -22.22%. 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. 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 3.

So we could say that we have a horizontal asymptote at ( y = 3 ). We could also—and there's a more rigorous way of defining it—say that our limit as ( x ) approaches infinity of the expression or of the function is equal to 3. Notice my mouse is covering a little bit, but as we get larger and larger, we're getting closer and closer to 3.

In fact, we're getting so close now that, well, here you can see it, we're getting closer and closer and closer to 3. You could also think about what happens as ( x ) approaches negative infinity. Here, you're getting closer and closer and closer to 3 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 ( f(x) ). 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 a key difference between a horizontal and a vertical asymptote. For 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.