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## AP®︎ Calculus AB (2017 edition)

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

Lesson 4: One-sided limits# One-sided limits from graphs

A one-sided limit is the value the function approaches as the x-values approach the limit from *one side only*. For example, f(x)=|x|/x returns -1 for negative numbers, 1 for positive numbers, and isn't defined for 0. The one-sided *right* limit of f at x=0 is 1, and the one-sided *left* limit at x=0 is -1. Created by Sal Khan.

## Want to join the conversation?

- I understand how quadratic equations, conic sections, and sine waves look on a graph and they're all very smooth. Is there ever a single equation that, when graphed, would look so broken and crooked as this first graph? Any of these graphs, for that matter.(28 votes)
- Do you mean graphs with discontinuities?

Many graphs that aren't piece-wise functions have discontinuities.

y=1/x

https://www.google.com/search?q=y%3D1%2Fx&rlz=1C1ASUT_enUS503US503&oq=y%3D1%2Fx&aqs=chrome.0.69i57j0l3j69i62l2.1166j0&sourceid=chrome&ie=UTF-8

Here, there is a discontinuity at x=0. From one side, the graph goes down to

-Infinity while from the other side, the graph goes up to Infinity. This is called an infinite discontinuity. Can you ascertain why?

y=x/x

https://www.google.com/search?q=y%3Dx%2Fx&rlz=1C1ASUT_enUS503US503&oq=y%3Dx%2Fx&aqs=chrome.0.69i57j0l3j69i62l2.1134j0&sourceid=chrome&ie=UTF-8

This is like the piece-wise function y=1 for x does not equal 0 (x/x=1) and y is undefined for x=0 (0/0 is indeterminate) (put your mouse art x=0 and see what y is). This is a removable discontinuity because all we did was remove one point from the graph and let the graph be normal everywhere else.

Let's say that ⌊x⌋ is the greatest integer less than or equal to x. Yes, this is a mathematical function.

y=x-⌊x⌋

I couldn't find a graph of this...

Let's try to visualize this. At every integer x, ⌊x⌋ is x. At every integer x,

x-⌊x⌋=x-x=0. However, from the left side, the function will be approaching 1. This is because the distance between two consecutive integers is 1. From the left side of integers, the numbers will be close to an integer, but the ⌊x⌋ of x will be all the way at the last integer. This means that x-⌊x⌋ is approaching x-(x-1), or 1. However, when it's supposed to get to 1, it jumps down to 0 every time. These discontinuities are called jump discontinuities because you jump from one place to another.

I hope this helps you understand discontinuities outside of piece-wise functions!(65 votes)

- why is a limit not defined if the left-handed limits and the right-handed limts are unequal??(22 votes)
- You could think about it like this: as you approach some point on the graph from opposite directions and you are approaching different y values, what would you say that the limit is? Is it one value? Is it the other? Well, it's both, depending on which direction you're coming from. So we say that there is no single 'ordinary' limit.(51 votes)

- How come the limit of f(x) is 5 in5:29. Shouldn't it be -5 because both limits from both directions are -5?(14 votes)
- You are correct! Good catch.

Also, notice as you watch the video at5:29that there is a small box in the lower right-hand corner of the video that says:

> Correction: The general limit is -5.

Have a great day! PS Minecraft is not so bad...(19 votes)

- This is hard. Im a year 8. Should I be learning this yet? I find it v. complicated and I dont understand it. The video helped a little. Thanks :)(4 votes)
- Calculus is usually presented in year 11 or year 12, though different countries have their own standards.

Before attempting calculus, you need to have mastered algebra, geometry and trigonometry. In calculus, it is assumed that you know those topics very well. If you haven't studied those topics, this material is going to be too advanced for you to understand.(42 votes)

- Why is limit the same whether the graph has an open circle [as in the endpoint is not included in the solution] and when it has a closed one? Thanks(6 votes)
- Are you familiar with Zeno's (ancient Greek philosopher) paradox? I find it useful to explain this concept of limits. Zeno argued that motion was impossible since, for example if you shot an arrow at a target the arrow would first have to travel half the distance to the target, then half the remaining distance, the half again and again and again, ad infinitum and thus never reach the target since the arrow always has to traverse the remaining half distance. Now in our physical world, that doesn't happen (and why is another discussion) BUT in the abstract math world, THIS IS TRUE. If you tell me you are x distance away from something, I can half the distance and be even closer than you, NO MATTER WHAT DISTANCE x you choose. With limits we are saying that no matter how close you want to get to the limit value, you can ALWAYS get closer - it doesn't matter if the limit value endpoint is included or not, you will never 'get' to it anyway since you can always half the distance your are from it. But since in either case (endpoint value included or not) you can keep getting as close as you want to it, the limit is the same. Hope that helps :)(18 votes)

- Is there a math symbol for 'does not exist'?(6 votes)
- I like to use the empty set symbol, which is a circle with a slash through it.(3 votes)

- What is actually meant by limit exist , i mean literally what is the significance when we say limit exists?(1 vote)
- Do you mean a rigorous mathematical definition or an explanation in simpler terms?

For the simple explanation:

A limit is said to exist for some function f(x) for some value c if f(x) clearly gets closer and closer to some finite value as x gets closer and closer to c. This means that you get the SAME value whether x is less than c and increasing toward c OR x is greater than c and and decreasing toward c. If all these conditions are met, the limit is said to exist. If any is not met, the limit is said to fail to exist or just "does not exist".

NOTE (and this is EXTREMELY important to understand): what the function is approaching as f(x) approaches x=c does NOT have to be the same thing as f(c). If there is a sudden change in f(x) at x=c, then the limit (if it exists) would be what f(x) "should have been" had the sudden change not been there. Also, a limit can SOMETIMES exist where the function itself is undefined.

As for the formal definition of a limit, I will defer to one of the professional mathematicians / math majors because that gets very tedious and I don't want to risk making a mistake.(10 votes)

- Can a function actually exist that reflects the above irregular graph? I believe functions usually shows some pattern instead of the sudden irregular graph like the above. please let me know.(3 votes)
- Certainly, we can make functions as irregular as we want as long as they don't violate the definition of a function. The function shown in the video is likely a piecewise function. If you want to learn more about piecewise functions, I have included the link to the khan academy video.

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:absolute-value-piecewise-functions/x2f8bb11595b61c86:piecewise-functions/v/piecewise-function-example(6 votes)

- If we have

f(x)=x,x<5,

then does the limit of x approaches 5 exists?(4 votes)- In order for a limit to exist, both one-sided limits must be equal. Since finding one of the one-sided limits at the endpoint of a function is impossible, the limit as a function approaches an endpoint does not exist. In your example, however, the limit of f(x) as x approaches 5 from the negative side does exist (and equals 5). Hope this helps!(3 votes)

- For the first graph, doesn't the limit f(x) as x approaches 4 = -5 and not positive 5?(3 votes)
- Yes, I noticed that too and Sal just made a mistake.(4 votes)

## Video transcript

So if we were to
ask ourselves, what is the value of our
function approaching-- as we approach x equals 2 from
values less than x equals 2. So as you imagine, as
we approach x equals 2-- So x equals 1, x equals 1.5,
x equals 1.9, x equals 1.999, x equals 1.99999999. What is f of x approaching? And we see that f of x
seems to be approaching this value right over here. It seems to be approaching 5. And so the way we
would denote that is the limit of f of x,
as x approaches 2-- and we're going to specify the
direction-- as x approaches 2 from the negative
direction-- we put the negative as a
superscript after the 2 to denote the direction
that we're approaching. This is not a negative 2. We're approaching 2 from
the negative direction. We're approaching 2
from values less than 2. We're getting closer and closer
to 2, but from below-- 1.9, 1.99, 1.99999 . As x gets closer and
closer from those values, what is f of x approaching? And we see here that
it is approaching 5. But what if we were asked the
natural other question-- What is the limit of f of
x as x approaches 2 from values greater than 2? So this is a little superscript
positive right over here. So now we're going to
approach x equals 2, but we're going to approach
it from this direction-- x equals 3, x equals 2.5, x
equals 2.1, x equals 2.01, x equals 2.0001. And we're going to get
closer and closer to 2, but we're coming from values
that are larger than 2. So here, when x equals
3, f of x is here. When x equals 2.5,
f of x is here. When x equals 2.01, f of x
looks like it's right over here. So in this situation, we're
getting closer and closer to f of x equaling 1. It never does quite equal that. It actually then just
has a jump discontinuity. This seems to be the limiting
value when we approach when we approach 2 from
values greater than 2. So this right over
here is equal to 1. And so when we think
about limits in general, the only way that a limit
at 2 will actually exist is if both of these
one-sided limits are actually the same thing. In this situation, they aren't. As we approach 2
from values below 2, the function seems
to be approaching 5. And as we approach 2
from values above 2, the function seems
to be approaching 1. So in this case, the limit--
let me write this down-- the limit of f of
x, as x approaches 2 from the negative
direction, does not equal the limit of f
of x, as x approaches 2 from the positive direction. And since this is the case--
that they're not equal-- the limit does not exist. The limit as x
approaches 2 in general of f of x-- so the limit of
f of x, as x approaches 2, does not exist. In order for it to have
existed, these two things would have had to have
been equal to each other. For example, if
someone were to say, what is the limit of f
of x as x approaches 4? Well, then we could think about
the two one-sided limits-- the one-sided limit from
below and the one-sided limit from above. So we could say,
well, let's see. The limit of f of x, as x
approaches 4 from below-- so let me draw that. So what we care
about-- x equals 4. As x equals 4 from below-- So when x equals 3,
we're here where f of 3 is negative 2. f of 3.5
seems to be right over here. f of 3.9 seems to be right
over here. f of 3.999-- we're getting closer and
closer to our function equaling negative 5. So the limit as we
approach 4 from below-- this one-sided
limit from the left, we could say-- this is going
to be equal to negative 5. And if we were to ask
ourselves the limit of f of x, as x approaches
4 from the right, from values larger than
4, well, same exercise. f of 5 gets us here. f of 4.5 seems
right around here. f of 4.1 seems right
about here. f of 4.01 seems right around here. And even f of 4 is
actually defined, but we're getting
closer and closer to it. And we see, once again,
we are approaching 5. Even if f of 4 was not
defined on either side, we would be
approaching negative 5. So this is also
approaching negative 5. And since the limit
from the left-hand side is equal to the limit
from the right-hand side, we can say-- so these
two things are equal. And because these
two things are equal, we know that the limit of
f of x, as x approaches 4, is equal to 5. Let's look at a
few more examples. So let's ask ourselves
the limit of f of x-- now, this is our new f of x depicted
here-- as x approaches 8. And let's approach
8 from the left. As x approaches 8 from
values less than 8. So what's this going
to be equal to? And I encourage you to pause the
video to try to figure it out yourself. So x is getting closer
and closer to 8. So if x is 7, f of 7 is here. If x is 7.5, f of 7.5 is here. So it looks like
our value of f of x is getting closer and
closer and closer to 3. So it looks like the limit
of f of x, as x approaches 8 from the negative
side, is equal to 3. What about from
the positive side? What about the
limit of f of x as x approaches 8 from the
positive direction or from the right side? Well, here we see as x
is 9, this is our f of x. As x is 8.5, this
is our f of 8.5. It seems like we're
approaching f of x equaling 1. So notice, these two
limits are different. So the non-one-sided limit,
or the two-sided limit, does not exist at f of
x or as we approach 8. So let me write that down. The limit of f of x,
as x approaches 8-- because these two things
are not the same value-- this does not exist. Let's do one more example. And here they're actually
asking us a question. The function f is graphed below. What appears to be the value of
the one-sided limit, the limit of f of x-- this is f of x--
as x approaches negative 2 from the negative direction? So this is the negative 2
from the negative direction. So we care what happens as
x approaches negative 2. We see f of x is actually
undefined right over there. But let's see what
happens as we approach from the negative
direction, or as we approach from values less
than negative 2, or as we approach from the left. As we approach from the
left, f of negative 4 is right over here. So this is f of negative 4. f of negative 3 is
right over here. f of negative 2.5 seems
to be right over here. We seem to be getting
closer and closer to f of x being equal
to 4, at least visually. So I would say that
it looks-- at least, graphically-- the limit of
f of x, as x approaches 2 from the negative
direction, is equal to 4. Now, if we also asked
ourselves the limit of f of x, as x
approaches negative 2 from the positive direction,
we would get a similar result. Now, we're going to
approach from when x is 0, f of x seems
to be right over here. When x is 1, f of x
is right over here. When x is negative
1, f of x is there. When x is negative 1.9, f of
x seems to be right over here. So once again, we seem to be
getting closer and closer to 4. Because the left-handed limit
and the right-handed limit are the same value. Because both one-sided limits
are approaching the same thing, we can say that the limit
of f of x, as x approaches negative 2-- and this
is from both directions. Since from both directions, we
get the same limiting value, we can say that the
limit exists there. And it is equal to 4.