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Calculus, all content (2017 edition)
Course: Calculus, all content (2017 edition) > Unit 1
Lesson 3: Limits from graphsLimits from graphs: asymptote
Sal finds the limit of a function given its graph. The function has an asymptote at the limiting value. This means the limit doesn't exist.
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And in general teachers and instructors what approach do they follow: the intuitive one (infinity) or traditional one (no limit)?(13 votes)
According to Stewart - Calculus_ Early Transcendentals 8th Edition, the infinite limit (f(x) approach infinity) means the limit doesn't exist but we would put infinity to describe its end behaviors.
The instructor may ask you
- Evaluate the limit. Then you would put infinity as the answer
- Does the limit exist? The answer would be no (because infinity is an arbitrary number that can be as large as we want).
Hope that helps.(70 votes)
So this is a weird question, but it's been bothering me...
When you evaluate a function to determine the limit, if the limit value is equal when approaching from both directions on the number line, then we say "the limits equal each other, therefore the limit exists AND has the value of ___ ."
However, if while approaching the variable from different directions and the limits do not equal each other or converge on infinity, then we say, "the limit does not exist."
But in the process of approaching the limit, even the ones we say have a congruent value and therefore exist, there is always an infinite amount of divisions while approaching the limit. If there are an infinite amount of steps, then how can we say with certainty that the limit exists? We grounded the very existence of limits in their having congruent values. However... you can make a function whereby you approach the same limit but by which the movement towards the limit do not have identical relations. This, in my mind, makes them not congruent. We also said that if the value of the limit is infinity, it's undefined. It's undefined because we cannot define infinity. Or that the totality of possible infinite series may have different sums and therefore there are an infinite amount of infinites ad infinitum. Why do we accept that to reach the limit there is an infinite amount of steps (which may or may not be congruent with eachother every step of the way, relative to direction) as having existence but don't accept the existence of limits that have as their value the very constituent by which we achieve the existence of the former?(7 votes)
We can say that the limit exists because we've shown in full generality that the function can get as close to the limit as we like. When we prove this using the definition, we are not "taking infinitely many divisions".
The limit of any continuous function can be approached infinitely many different ways. The limit as x goes to 0 of f(x)=x can be approached by repeatedly halving the distance to 0, or quartering it, or any other division. In your mind, does that mean that f(x)=x can take on infinitely many different values at 0? The fact that it doesn't matter how x approaches 0 is why we can define a limit; we say we want the function to get within 𝜀 of some value f(c), and we know that we can find an x within ∂ of c. It doesn't matter what order we choose the x's, if we're taking a bunch. We can always do it for any of them.
When we say that a limit goes to infinity, we are not saying the value of the limit is infinity. Writing "lim f(x)= ∞" is shorthand for saying that the function gets arbitrarily large, that for any value the function takes on, we can find a spot where it's even larger, and larger by any amount. So the function does not "approach" any single real number. That's why the limit is undefined. It's not because we can't define infinity. (Our grasp of infinity has been solid and getting better since 1891.)
I'm not sure what you're trying to say about infinite series, but there are an infinite number of infinities. If you would clarify that part, I'd love to help.(5 votes)
how do you reach infinity(2 votes)- You don't. You can only approach it. Once I've reached a number that seems infinitely large, like say 10^10^10^10^10^10^10, there is a number that is even bigger than that.(5 votes)
I thought infinity is a concept, not a number?(1 vote)- It is. When we say that a function goes to infinity, this is shorthand for saying that it gets arbitrarily large. For any value the function reaches, we can find a place where it reaches an even higher value.(5 votes)
so basically an asymptote is a continuous discontinuity(3 votes)
So is a limit for an asymptote always infinity or DNE(2 votes)
The limit could also be negative infinity depending on which direction your function is heading.(1 vote)
- In the graph,how to draw the line, do you draw it based on the information or make some random lines?(3 votes)
Based on the information if you are asked to "graph it", draw something but label it correctly and accordingly when you are asked to "sketch it"(1 vote)
- Wouldn't saying that the limit is infinity be treating infinity as a numerical value, which, by definition, is not a numerical value?(1 vote)
- When you say "the limit is infinity", it is basically a simplification of what you actually mean. What you actually mean is that "the limit tends towards infinity". It's basically a "slang" term that is accepted because the intended meaning is understood.(2 votes)
What is the actual definition of asymptote? It talks about it but doesn't say what it is.(0 votes)- An asymptote is a line that does not touch or intersect the function, but gets arbitrarily close to it.(3 votes)
please i did not really understand it(1 vote)- Its one of the continuities, if you have a problem, here is a summary:
https://www.youtube.com/watch?v=YmuqoXnCSNI(1 vote)
Video transcript
- [Voiceover] Alright we have a graph of y is equal to f of x and we wanna figure out
what is the limit of f of x as x approaches negative three. And if we just look at
x equals negative three, it's really hard to see at least based on how this graph looks, how what f of negative
three is and if anything, it looks like we have an
asymptotic discontinuity here. It looks like on the left side
of x equals negative three, it looks like we are approaching, I guess you can say infinity and on the right side, it looks like we are
approaching infinity as well. And we can just look at that and say, What is f of negative 5? That's four. F of negative four is
looks like it is around, I don't know around eight. F of negative three is off the charts. F of negative, if we continued with this trend
and if we were to asymptote towards this line right over here, this vertical asymptote, it looks like as we get closer
and closer to negative three that the value of the function
at that point is approaching, is getting closer and closer to infinity, at least that's what it looks like from what we can see on this graph as we approach negative three
from the left hand side. Let's think about as we
approach negative three from the right hand side, so this is f of negative one, f of negative two, f of negative 2.5 looks like
it's up here some place, f of negative 2.9 would be even higher, f of negative 2.999
looks like it will just once again approach infinity so this type of limit in some context you'd say this limit doesn't exist doesn't doesn't exist in the formal sense so that's one way to think about it. In some contexts, you'll hear people say that this limit since from the left and from the right, it looks like it's going to infinity. Sometimes you'll see people say that it's approaching infinity and so this is depending
on what type of a class what type of a context you're in but in the traditional sense of the limit or in the technical sense, there are ways that you can define limits where this will make a
little bit more sense but the traditional definition of a limit, it would be, you would say that this
limit does not exist.