Infinite limits and asymptotes

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 2 over X minus 1 so let me get that one graphed and so you can immediately see that something interesting happens at X is equal to 1 if you were to just substitute x equals 1 into this expression you're going to get 2 over 0 and whenever you get a non zero thing over 0 that's a good sign that you might be dealing with with a vertical asymptote in fact we can draw that vertical asymptote right over here at x equals 1 but let's think about how that relates to limits what if we were to explore the limit as X approaches 1 of f of X is equal to 2 over X minus 1 and we can think about it from the left and from the right so if we approach 1 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 0 the f of X would equal to Nate would be equal to negative 2 what X is equal to 0.5 f of X is equal to negative 4 and then it just gets more and more negative the closer we get to 1 from the left I could really so I'm not even that close yet if I get to let's say point 9 1 I'm still 9 hundredths less than 1 I'm at negative 22 point 2 2 2 already and so the limit as we approach 1 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 tour 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 3 and so we could say that we have a horizontal asymptote at y is equal to 3 and we could also and there's a more rigorous way of defining it saying that say that our limit as X approaches infinity is equal of the expression or of the function is equal to 3 notice my mouse is covering it a little bit as we get larger and larger we're getting closer and closer to 3 in fact it's we're getting so close now that well here you can see we're getting closer and closer and closer to 3 and you can also think about what happens as X approaches negative infinity and 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 sine of X and so there you have it we are now oscillating around the horizontal asymptotes 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 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 gets goes to positive infinity or as X goes to negative infinity