Calculus, all content (2017 edition)
Limits from graphs: limit isn't equal to the function's value
Sal finds the limit of a function given its graph. The function's value at the limit is different from the limit's value, but that doesn't mean the limit doesn't exist!
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- What if the point of discontinuity falls on the function? Does that mean it fills a hole or did it just happen to fall there?(12 votes)
- If that were the case then it wouldn't be a point of discontinuity. It would simply be a continuous function.(44 votes)
- At1:00Sal talks of "g of negative 6.999." Does he mean to say "negative?" Isn't the 6.999 positive?(5 votes)
- He said "g of negative 6.999" but it should be "g of 6.999". I'm guessing what he was trying to say that g of 6.999 is still negative but got mixed up. Report it so he can add an annotation for correction.(9 votes)
- I don't understand the point here? You have graphed an undefined function then picked a random point off the line and are showing that it is off the line... ok
It would be very helpful here to give us the function that plots out to the graph shown yet is discontinuous at X=7(4 votes)
- He made it much easier for us by showing us only the line. It is much simpler and easier to understand than if he actually defined the function.(2 votes)
- Would it be possible to actually write up a function for this graph with the discontinuity, or is it purely hypothetical?(3 votes)
- You can define such a function as a "piecewise continuous" function. For certain parts of the graph, it'll act like one function, but on other parts, it'll act in a completely different way.(6 votes)
- Is the following statement correct: The function approaching 7 is zero, however, the function at 7 is 3. Thanks for any feedback!(3 votes)
- what is the function g(x) defined as?(2 votes)
- He does not define the function, as far as I could see.(2 votes)
- Why the function's actual value g(7) is not greater than 9?(1 vote)
- The little blue dot that is disconnected from the curve is where g(7) is defined, which as a value of 3, which is less than 9.(4 votes)
- But in all the previous cases we sae that the limit was approaching infinity but the value of limit wasnt infinity ... So is there discontinuity present too?(2 votes)
- Are we supposed to be already aware of what "Continuity" is; before we begin learning this (Calculus) subject?! ^_^(1 vote)
- It is possible to learn how to compute integrals and derivatives without understanding continuity.
But if you really want to understand what calculus is, you'll start with some basic notions like continuity (or even more basic things like limits of sequences). You can then build up from there, so even when you get to more advanced parts of calculus, you can still understand it in terms of these basic notions.(2 votes)
- Is it possible to figure out the limit of functions without tables or graphs??(1 vote)
- Yes e.g.1:
y = x (linear function, continuous)
thus the limit of any point can be obtained by subbing in the x value into the equation.
The other two ways are factoring and multiply by the conjugate, if you have time, check out this video:
- [Voiceover] So, here we have the graph y is equal to g of x. We have a little point discontinuity right over here at x is equal to seven, and what we want to do is figure out what is the limit of g of x as x approaches seven. So, since you say, well, what is the function approaching as the inputs in the function are approaching seven? So, let's see, so if we input as the input to the function approaches seven from values less than seven, so if x is three, g of three is here, g of three is right there, g of four is right there, g of five is right there, g of six looks like it's a little bit more than, or a little bit less than negative one, g of 6.5 looks like it's around negative half, g of negative 6.9 is right over there, looks like it's a little bit less than zero, g of negative 6.999 looks like it's a little bit, it's still less than zero but it's a little bit closer to zero so it looks like we're getting closer as x gets closer and closer but not quite at seven, it looks like the value of our function is approaching zero. Let's see if that's also true from values for x values greater than seven, so g of nine is up here, looks like it's around six, g of eight looks like it's a little bit more than two, g of 7.5 looks like it's a little bit more than one, g of 7.1 looks like it's a little bit more than zero, the g of 7.1 looks like it's a little bit more than zero, g of 7.01 is even closer to zero, g of 7.0000001 looks like it'll be even closer to zero so once again it looks like we are approaching zero as x as approaches seven, in this case as we approach from larger values of seven. And this is interesting because the limit as x approaches seven of g of x is different than the function's actual value, g of seven, when we actually input seven into the function. When we actually input seven into the function, we can see the graph tells us that the value of the function is equal to three. So we actually have this point discontinuity, or sometimes called a removable discontinuity, right over here, and this is, I'm not gonna do a lot of depth here, but this is starting to touch on how we, one of the ways that we can actually test for continuity is if the limit as we approach a value is not the same as the actual value of the function of that point, well, then we're probably talking about, or actually we are talking about, a discontinuity.