AP®︎ Calculus BC (2017 edition)
This video explores estimating one-sided limit values from graphs. As x approaches 6 from the left, the function becomes unbounded with an asymptote, making the left-sided limit nonexistent. However, when approaching 6 from the right, the function approaches -3, indicating that the right-handed limit exists. Sal's analysis highlights the importance of understanding limits from both sides.
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- Does this mean that the general limit of g(x) as x approaches 6 does not exist? In previous videos Sal stated that if both one-sided limits are not equal, the general limit does not exist. But is it meaningful in this case to say that the one-sided limits are not equal, when one of them does not exist?(8 votes)
- The general limit as x approaches 6 does not exist. The limit as x approaches from the left is positive infinity, and thus does not exist. The limit as x approaches 6 from the right is -3.(18 votes)
- My question may seem dull at first, however I couldn't find an answer to it.
Based on my understanding on limits the main function of a limit is to find the near approximated output value when the function is not defined at that output. But what if the function was defined at that point, will the limit become useless?
In other words, are limits useful only at undefined points?(10 votes)
- see this video:
In this video Sal shows the limit at a point of a jump discontinuity. The function IS defined at the point of discontinuity. And the limit is a different value than the value of the function.
I am not sure if that video will answer your question. If you are wondering what would happen if you were to find the limit on a continuous function the limit would just be the value of the function at that point.(14 votes)
- what are limits exactly used for ?(11 votes)
- Calculus limits have a wide range of applications in various fields. Engineers rely on limits for designing structures, analyzing circuits, optimizing systems, and solving differential equations. In finance, limits are used to calculate interest rates, evaluate investments, and assess risk and probability. They are also employed in pharmacokinetics to determine optimal dosing. A fun example is tracking aircraft: by collecting position data points over a short time interval and taking the limit as the interval approaches zero, we can calculate the aircraft's instantaneous velocity at any given moment. In summary, calculus limits are incredibly versatile and find applications in numerous areas, from engineering and finance to medicine and even more.(2 votes)
- So, the limit of g(x)as x →6 doesn't exist, right ?(7 votes)
- Correct, because the limit from the negative side is not equal to the limit from the positive side (and because the limit does not exist for the asymptote)(8 votes)
- Can I say its unbounded, instead of saying "it does not exist"?(6 votes)
- Yes, you can use the term "unbounded" to describe the behavior of the function g(x) as x approaches six from the left, instead of saying "it does not exist." Saying that the limit is unbounded means that the function grows without bound and becomes infinitely large as x approaches six from the left. Both "unbounded" and "does not exist" convey the same idea in this context.(5 votes)
- What kind of functions acutally produces this kind of "unbounded" limits?(4 votes)
- I am still a little confused about the limit of g(x) as "x" approaches 6 from the left. How come that limit does not exist, and is not equal to infinity?(2 votes)
- Infinity is not a number, it is a concept. We use it to concisely say that we're approaching something but never being able to attain it, and are describing the manner in which it does so. As I get closer and closer to 6 from the left, I get higher and higher in y-value. There is no y-value such that x will actually "touch" 6 from the left, so the limit does not exist.(9 votes)
- I'm sorry but shouldn't Sal have answer the limit of f(x) as x---> 6 from the left side, shouldn't the answer be infinity? why did he write does not exist(I understand it doesn't because both sides are not equal, but if we look at it in more details technically it doesn't exist but from the left side it should be equal to infinity and from the right side is equal to -3(5 votes)
- correct. because he put the "-" next to the 6, it means that it is coming in from a value less than x=6. If he had not put it, then he would be correct because without a "+" or "-" then it means across the entire graph, not just a greater than or less than value of x=6.(1 vote)
- What's the difference between an undefined limit and a limit that doesn't exist?(3 votes)
- They're the same. A limit is defined as a real number L that satisfies the epsilon-delta definition. If there is no such real number, then saying that such a number doesn't exist is the same as saying it's undefined.(4 votes)
- When you say asymptote in the video description, what do you mean, because my(apparently faulty) understanding was that it was just the line where something started or approached or something like that. Also, is there any form of calculus in which exponential growth and decay equations are important. And finally, sorry for all the questions, but how is this useful or will apply to life, besides being a fun way to pass the time, I mean.(2 votes)
- A vertical asymptote is a vertical line that a graph will steadily approach as it gets closer and closer to where the x value. A horizontal asumptote meanwhile is a horizontal line the graph gets closer to as the graph headsto an infinity. A good simple graph that demonstrates both is 1/x, with a horizontal asymptote at y=0 and a vertcal asymptote at x=0.
I know logarithmic growth is a topic in differential equations.
And aside from jobs that specifically use these skills, the problem solving in general is useful, and if you ever model something with a graph calculus would be able to easily tell you maxs, mins, and how fast things are changing, which can be useful for speeds. Again, these can be of varying importance, but it will be nice to know if you need to.(4 votes)
- [Voiceover] Over here we have the graph of y is equal to g of x. What I wanna do is I wanna figure out the limit of g of x as x approaches positive six from values that are less than positive six or you could say from the left, from the, you could say the negative direction. So what is this going to be equal to? And if you have a sense of it, pause the video and give a go at it. Well, to think about this, let's just take different x-values that approach six from the left and look at what the values of the function are. So g of two, looks like it's a little bit more than one. G of three, it's a little bit more than that. G of four, looks like it's a little under two. G of five, it looks like it's around three. G of 5.5, looks like it's around five. G of, let's say 5.75, looks like it's like nine. And so, as x gets closer and closer to six from the left, it looks like the value of our function becomes unbounded, it's just getting infinitely large. And so in some context, you might see someone write that, maybe this is equal to infinity. But infinity isn't, we're not talking about a specific number. If we're talking technically about limits the way that we've looked at it, what is, you'll sometimes see this in some classes. But in this context, especially on the exercises on Khan Academy, we'll say that this does not exist. Not exist. This thing right over here is unbounded. Now this is interesting because the left-handed limit here doesn't exist, but the right-handed limit does. If I were to say the limit of g of x as x approaches six from the right-hand side, well, let's see. We have g of eight is there, go of seven is there, g of 6.5, looks like it's a little less than negative three. G of 6.01, little even closer to negative three. G of 6.0000001 is very close to negative three. So it looks like this limit right over here, at least looking at it graphically, it looks like when we approach six from the right, looks like the function is approaching negative three. But from the left, it's just unbounded, so we'll say it doesn't exist.