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### Course: Differential Calculus>Unit 1

Lesson 14: Infinite limits

# Infinite limits and asymptotes

Unbounded limits are represented graphically by vertical asymptotes and limits at infinity are represented graphically by horizontal asymptotes.

## Want to join the conversation?

• That wavy transformation using sin(x) was pretty cool.
• I thought you could never cross asymptotes, only approach; if so then you could definitely cross vertical asymptotes, like the multy-function 3y/(y^2-5)+sin(y)=x
• Here is the confusing thing about asymptotes. You can never cross a vertical asymptote, but you can cross a horizontal or oblique (slant) asymptote. The reason you cannot cross a vertical asymptote is that at the points on the asymptote, the function is undefined because the x value would make the denominator zero. I hope this makes sense!
• Can there be diagonal asymptotes?
• The horizontal cases seem a bit counter-intuitive for me. Why do we talk about it here. Are there any applications of the horizontal asymptotes in maths problems or in real life. I just felt a bit confused about graphs with horizontal asymptotes.
• Plenty of applications. A horizontal asymptote can often be interpreted as an upper or lower limit for a problem. For example, if we were to have a logistic function modeling the spread of the coronavirus, the upper horizontal asymptote (limit as x goes to positive infinity) would probably be the size of the Earth's population, since the maximum number of people that can get the virus is the number of people on the planet. If you are not familiar with the shape of a logistic graph please look it up to see.

Hyperbolic functions such as y = 1/x represent an inverse variation relationship between two quantities x and y. Many things in life vary inversely, for example the number of cigarettes someone smokes and their life expectancy. If you smoke 10 packs a day, your life expectancy will significantly decrease. The horizontal asymptote represents the idea that if you were to smoke more and more packs of cigarettes, your life expectancy would be decreasing. If it made sense to smoke infinite cigarettes, your life expectancy would be zero.
• why wont a vertical asymptote be crossed ? ( said in )
• a vertical asymptote could have a value for 1 point if you use a piecewise function, but it won't be crossed several times because that would result in several y-values for one x-value, which is not allowed for functions.
• How can we distinguish a horizontal and vertical asymptote in higher dimensions? Don't they loose meaning? Since everything is according to once point of reference (this might be the key, BUT) Does that mean that for every object in space we asign a coordinate plane to determine their point and asymptotes?

Sorry if I cannot get to explain myself or this may sound stupid, so many questions.
• We tend not to use the terms 'horizontal/vertical asymptote' in more than 2 dimensions. If a curve or surface was asymptotic to some line or plane, we would care more about finding the equation of that line or plane, which tells us everything we want to know about it, than about whether to call it horizontal or vertical.
(1 vote)
• i do not understand what are asymptotes. Is there nother video that explains it or can someone help ?

thank you
• An asymptote is when the line approaches an x or y value, but does not reach it.
To get a visual on this topic, I would plug the equation y=1/x into a graphing calculator. The asymptotes that you will see are x=0, (the line soars up to infinity on one side, and down to negative infinity on the other), and y=0, (as x goes to infinity, the line gets closer and closer to the x-axis, but it never touches).

I also tried to find a video on this topic, but I couldn't find one, so I hope my explanation helps you out.
• If I want to get limit of f(x) as x approaches to infinity, how do we get that limit from positive side? It seemed that if this works, there has to have something bigger than infinity
(1 vote)
• SAL finaslly used desmos calculator les gooooooooooooo
• But essentially isn't this all speculation? we look at the end behavior of a graph and assume it will continue to approach a certain value. how do we know for sure? we can prove many different patterns, but how do we prove one like this?

also, another question on the topic of speculation. what if a graph's slope is so so so infinitesimally small that we can never "get" to the interval where the function appears to be approaching a certain value because, again, it is physically not possible for us to see.
(1 vote)
• The definition of an asymptote is a line that continually approaches a given curve but never reaches it. If it continually approaches the curve, it will never do anything "weird" as it approaches infinity, otherwise we would detect this in the limit process. Even without a picture or graph, you can determine many things about a function just from numbers and algebra because the picture of a function is just the result of how the function behaves algebraically. In other words, the picture does what the function does, so if we determine how the function behaves, the graph will just tell us the same thing pictorially.

In regards to your second question, one of the beautiful things about math is that we don't always need to see in order to believe. It is true that we cannot physically observe an infinitesimally small slope or what a function does as it approaches infinity, but functions have certain behavior modeled by the structure they are given when they are defined. For example, if a function is said to exist in the real numbers, then because the real numbers are a set with all sorts of structure and properties, the function will inherit those behaviors and properties (most of the time, unless you're doing something weird with your function). This is mostly gibberish to say that if a function behaves a certain way indefinitely, then when the function reaches "indefinitely", we just say that it's the value it was approaching. For example, y = 1/x. As x approaches infinity, the limit of y gets so stinking close to 0 that we just call it 0 because that's where y looks like its going (limit essentially asks "what does y look like it's approaching?).

## Video transcript

- [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.