How limit is defined, If there are two points on f(x) ?

Name: Limits from graphs: non-integer limit
Uploaded: 2021-07-03T00:14:30Z
Description: When reading limits from graphs, the limit can sometimes be a fraction. No reason to panic! Keep calm and approximate the limit.

Sal goes over this multiple times, and if you are still confused after my explanation i advise to go check out previous videos of AP Calc AB- Limits, but the idea is that a Filled-in dot is a confirmed value, where-as a hollow dot is a Hole, or, to put it better, the function is undefined at that value. So if we look at the graph, we see that as X Approaches -7, the values on either side of X=-7 get closer and closer to a specific value, yet never quite reach it. This is the "Limit" of the function as X approaches -7 because it can never quite reach that value, but it is continouslly getting closer and closer to it. On the other hand, the filled in dot is the confirmed value of F(X) or the function when X=-7. So if the question had asked instead of the limit of F(x) what the value of F(X) when X=-7 was the answer would be -4. it's confusing but in summary, a limit's value at X depends on the graph's trend or, more accurately, the points around it, while the value of a Function at X depends on the confirmed value of that function.

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Course: Staging content lifeboat > Unit 11

Lesson 1: Internal

Limits from graphs: non-integer limit

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When reading limits from graphs, the limit can sometimes be a fraction. No reason to panic! Keep calm and approximate the limit.

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Lakshaydeep Chaudhary
Posted 7 years ago. Direct link to Lakshaydeep Chaudhary's post “How limit is defined, If ...”
How limit is defined, If there are two points on f(x) ?
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(1 vote)
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- Sam Dumesh
  Posted 7 years ago. Direct link to Sam Dumesh's post “Sal goes over this multip...”
  Sal goes over this multiple times, and if you are still confused after my explanation i advise to go check out previous videos of AP Calc AB- Limits, but the idea is that a Filled-in dot is a confirmed value, where-as a hollow dot is a Hole, or, to put it better, the function is undefined at that value.
  
  So if we look at the graph, we see that as X Approaches -7, the values on either side of X=-7 get closer and closer to a specific value, yet never quite reach it. This is the "Limit" of the function as X approaches -7 because it can never quite reach that value, but it is continouslly getting closer and closer to it.
  
  On the other hand, the filled in dot is the confirmed value of F(X) or the function when X=-7. So if the question had asked instead of the limit of F(x) what the value of F(X) when X=-7 was the answer would be -4.
  
  it's confusing but in summary, a limit's value at X depends on the graph's trend or, more accurately, the points around it, while the value of a Function at X depends on the confirmed value of that function.
  Button navigates to signup page
  (2 votes)

Video transcript

- [Instructor] The function h is defined for all real numbers in the graph y is equal to h of x, right over here. That's what they're showing us. And they ask us, what is a reasonable estimate for the limit as x approaches negative seven of h of x, and they give us some choices for those reasonable estimates including the possibility that the limit does not exist. So encourage you to pause the video and see if you can work through this yourself. All right, now let's think about what's going on. So something is interesting at x equals negative seven. At x equals negative seven, we have the function. It is defined, but it is not continuous with the rest of the function. H of negative seven, h of negative seven is, looks like a little bit, it looks like it's like negative four point something. So it's approximately negative four point, I don't know, looks like negative 4.1 something, something, something. So that's what h of negative seven is. And sometimes there's this temptation to say, oh, whatever the function equals, that must be what the limit is. But this is actually a really good example for showing the difference, that many times, the limit exists and is approaching or it is a value different than what the value that the function takes on at that point. And so let's think about the limit. As x approaches negative seven from smaller values of x, so as x gets closer and closer to negative seven, it looks like the value of our function is approaching where we have this gap. So it looks like our function is getting closer, the value of our function is approaching this value right over here. Similarly, if we were to approach from the left, so for approach from more negative values, it looks like the value of our function is approaching that same thing. And since it looks like it's approaching the same value whether we're coming from more negative values or less negative values, we know, or we can assume, or we have a good sense that this limit exists. We seem to be approaching the same value. Now, it's a value different than the actual value of the function there. The value of the function there is at negative 4.1 or something like that. Now, this looks like the function itself is approaching, this looks like, I don't, one point, 1.2, 1.3, something like that. So now, let's look at our choices. What's a reasonable estimate for this limit? Well, it's definitely not negative seven. The function is definitely not approaching y equals negative seven. This is kind of a distractor choice because this is what x is approaching. X is approaching negative seven, but we want to know what is the function approaching? What is the value of h approaching as x approaches negative seven? So let's rule that one out. Negative 4.1, so negative 4.1 is an interesting choice because that's a good approximation for what the actual value of the function is there. The function is defined there, although it kind of jumps and is defined right over here, and it's discontinuous. So this is more of what h, this could be a reasonable estimate for h of negative seven, what the function actually is defined to be at x equals negative seven. But as we approach x equals, as we approach negative seven, it is not approaching, the value of the function does not seem to be approaching negative 4.1. So I'm gonna rule that one out. Now, 1.3, that is what, that is a pretty reasonable estimate for what the function seems to be approaching as x gets closer and closer to negative seven 'cause, you know, 6.9, 6.99, or negative 6.9, negative 6.99, negative 6.9999, from the right or from the left, negative 7.1, negative 7.01, negative 7.001, so on and so forth. It looks like the value of that function is approaching approximately 1.3. So I like that choice right over there. 1.8, well, that's just another choice but this, looking, approximating where this is, it definitely looks closer to one than it does to two. So I'd rule that one out. And I would say that this limit does exist because it looks like we're approaching the same value whether we're coming from below or from above, so to speak.