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### Course: AP®︎ Calculus AB (2017 edition)>Unit 1

Lesson 4: One-sided limits

# Connecting limits and graphical behavior

Usually when we analyze a function's limits from its graph, we are looking at the more "interesting" points. It's important to remember that you can talk about the function's value at any point. Also, a description of a limit can apply to multiple different functions.

## Want to join the conversation?

• This may be a bit stupid, but how do you find the limit as x approaches a specific value of a vertical line?
(35 votes)
• A vertical line is not a function so you can't take a limit involving one.
(108 votes)
• okay so i just did the practice test and it asked me to choose all that applies. It gave me a notation I haven't seen before which was a superscript of a + and a - sign. The lessons never mentioned those things. So I guess now I know that a negative superscript means it's coming from the left and a positive one means it's coming from the right.
(5 votes)
• The test questions show an additional positive or negative sign in some of the limits: x→−2− What does this mean??
​​
(4 votes)
• How limit of f(x) as x approaches pi is possible ? Can we find out limit of an irrational number ?
I believe we must be considering some approximate value like 3.14 etc.
(4 votes)
• It sure is possible! The fact that pi is a irrational number doesn't affect the limit/function in anyway. Graphically, of course, you can't tell the limit of f(x) when x approaches pi with accuracy, but if, instead, there was an expression for f(x) you could easily solve the limit and get an accurate number. If the limit of f(x) when x approaches pi was impossible just because pi is an irrational number, there would be infinite impossible limits in this function, since f(x) is defined by real numbers! And that's not true, since pretty much all limits, no matter the value to which x is approaching, are possible in this function.
Hope this helps!
(9 votes)
• The practice was one of the most frustrating things I have to endure.
(6 votes)
• Try solving for the position of an electron
(3 votes)
• How can we know whether a one-sided limit is the correct answer, or when a two-sided limit is needed?
Waldemar Klassen
(4 votes)
• You should always look for a 2 two-sided limit if applicable. By definition, a limit can exist only when the left and right-hand sides are equal. One-sided limits are usually used for checking continuity or determining behavior of the function.
(6 votes)
• How do you find the limit x approaches a specific value of a vertical line?
(2 votes)
• Those limits do not exist as, in a vertical line, there is a value of x for which you have infinite y values.
(3 votes)
• Is it possible to find the limit of a complex number ?
(2 votes)
• Yes. Carefully defining and understanding what that means, and exploring the consequences, leads to the branch of math called complex analysis.
(3 votes)
• What happens when the graph is undefined for a huge part around the point you want to find the limit for? Like if the green graph he drew in the video at was the finished graph what would the limit as x approaches 5 be?
(2 votes)
• If I understand you correctly, you are asking what the limit of that green graph would be as x approaches? Specifically, how the graph looks at is the graph we use??

If so:

The limit of this graph as x approaches 5, does not exist. It does not exist because the graph itself does not touch 5.

As in:
You can set x=5 in the function of the green graph and solve, but it would be undefined, and if you can't define a point of a graph, you can't define the limit around that point.

Remember that a graph of a function is a line consisting all the infinitely small points in which inputting the x-value of any of those points will give you an output of that y-value.
(3 votes)
• What if there is a piecewise function comprised of two functions that are undefined for K but the whole function approaches the same corresponding value for K from both sides?
Say f(x) = {x if x > 0, (x^2)/2 if x < 0}
f(x) approaches 0 as x does.
(2 votes)
• by definiton the limit will equal 0.
(2 votes)

## Video transcript

- [Instructor] So we have the graph of y is equal to g of x right over here. And I wanna think about what is the limit as x approaches five of g of x? Well we've done this multiple times. Let's think about what g of x approaches as x approaches five from the left. g of x is approaching negative six. As x approaches five from the right, g of x looks like it's approaching negative six. So a reasonable estimate based on looking at this graph is that as x approaches five, g of x is approaching negative six. And it's worth noting that that's not what g of five is. g of five is a different value. But the whole point of this video is to appreciate all that a limit does. A limit only describes the behavior of a function as it approaches a point. It doesn't tell us exactly what's happening at that point, what g of five is, and it doesn't tell us much about the rest of the function, about the rest of the graph. For example, I could construct many different functions for which the limit as x approaches five is equal to negative six, and they would look very different from g of x. For example, I could say the limit of f of x as x approaches five is equal to negative six, and I can construct an f of x that does this that looks very different than g of x. In fact if you're up for it, pause this video and see if you can so the same, if you have some graph paper, or even just sketch it. Well the key thing is that the behavior of the function as x approaches five from both sides, from the left and the right, it has to be approaching negative six. So for example, a function that looks like this, so let me draw f of x, an f of x that looks like this, and is even defined right over there, and then does something like this. That would work. As we approach from the left, we're approaching negative six. As we approach from the right, we approaching negative six. You could have a function like this, let's say the limit, let's call it h of x, as x approaches five is equal to negative six. You could have a function like this, maybe it's defined up to there, then it's you have a circle there, and then it keeps going. Maybe it's not defined at all for any of these values, and then maybe down here it is defined for all x values greater than or equal to four and it just goes right through negative six. So notice, all of these, all of these functions as x approaches five, they all have the limit defined and it's equal to negative six, but these functions all look very very very different. Now another thing to appreciate is for a given function, and let me delete these. Oftentimes we're asked to find the limits as x approaches some type of an interesting value. So for example, x approaches five, five is interesting right over here because we have this point discontinuity. But you could take the limit on an infinite number of points for this function right over here. You could say the limit of g of x as x approaches, not x equals, as x approaches, one, what would that be? Pause the video and try to figure it out. Well let's see, as x approaches one from the left-hand side, it looks like we are approaching this value here. And as x approaches one from the right-hand side, it looks like we are approaching that value there. So that would be equal to g of one. That is equal to g of one based on that would be a reasonable, that's a reasonable conclusion to make looking at this graph. And if we were to estimate that g of one is, looks like it's approximately negative 5.1 or 5.2, negative 5.1. We could find the limit of g of x as x approaches pi. So pi is right around there. As x approaches pi from the left, we're approaching that value which just looks actually pretty close to the one we just thought about. As we approach from the right, we're approaching that value. And once again in this case, this is gonna be equal to g of pi. We don't have any interesting discontinuities there or anything like that. So there's two big takeaways here. You can construct many different functions that would have the same limit at a point, and for a given function, you can take the limit at many different points, in fact an infinite number of different points. And it's important to point that out, no pun intended, because oftentimes we get used to seeing limits only at points where something strange seems to be happening.