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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition) > Unit 1

Lesson 4: One-sided limits- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- One-sided limits from graphs
- One-sided limits from tables
- 1-sided vs. 2-sided limits (graphical)
- Limits of piecewise functions: absolute value
- Connecting limits and graphical behavior (more examples)

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# Connecting limits and graphical behavior (more examples)

Sal analyzes various 1- and 2-sided limits of a function given graphically. Created by Sal Khan.

## Want to join the conversation?

- What is the point of taking a limit? What would be its application to everyday life? I seem to understand how it all works mathematically, but I don't understand WHY we use them.(105 votes)
- As someone who never liked maths much in high school, I can totally understand where you're coming from - but now as an adult studying maths at university I can affirm that pretty much everything you learn in maths is relevant, probably in ways you've never though of.

For example at our institution we are using math to do in silico trials. These are medical trials performed by a computer using software to run our equations to emulate/simulate a real trial. Using derivatives and limits we can look at how tumors grow in a particular type of cancer, to better understand the relationship between our immune system and cancer cells.(47 votes)

- What's the difference between 1) limit does not exists

2) limit undefined(27 votes)- You say does not exist when dealing with the limit.(ex:lim f(x))

You say undefined when dealing with a function alone(ex:f(x))

At least that is how my calculus teacher explains it.(10 votes)

- At1:05, when the tutor was explaining why it is false, can someone give a bit more of a explanation on why it is false cause I'm looking at the graph and I keep coming up with zero.(5 votes)
- I don't believe Sal actually made a mistake. He states that we are coming from the positive side, so we would approach from numbers greater than 1. At those numbers it appears that we approach 1, as he stated. If we had approached from the negative side then yes, we would be approaching 0. I believe you mistook the positive sign as a sign to go up the number line. That would explain why you are coming from the negative side.(9 votes)

- does the last example also mean that "limit exists at 1 as x approaches to 1.5?"

I mean if there is no sign to decide which side we approach from, It asks whether limit exists(so we should check from both sides)?(2 votes)- Yes, you should check from both sides.(1 vote)

- Shouldn't the limit of f(x) as x approaches 1 = 1 because the bubble is filled in for the straight line and not for the curved line?(4 votes)
- That is when you are coming from the right side and not the left.(1 vote)

- I m struggling with the precise definition of a limit (f(x) - L) < e and can't find a video which helps with this topic. Could anyone help out?(2 votes)
- So the limit as x approaches 0 in this function is 1? Even though we already have a clearly defined y value of "0" for x=0? Does that mean that limits don't take into consideration the actual values for when they reach their "destination"? I'm a little confused.(2 votes)
- The limit as you approach a particular value is what it looks like the function ought to be, which may or may not be what the function actually is at that point. Thus, if the function is continuous, the limit and the function will have the same value. But, if the function is not continuous at that point, there is a sudden change at that point, then the limit will be whatever the function would have been had there been no sudden change.(3 votes)

- Why is it called calculus? I don't understand that.(2 votes)
- "Calculus" is derived from the latin word meaning pebble, because that is what was originally used for calculations.(2 votes)

- If a limit can only be approached from one side such as lim(x->-1) arccos(x) does the limit exist?(2 votes)
- The one sided limit exists, but the 2 sided limit may or may not exist.

For your particular question, the lim(x->-1) arccos(x) does not exist because the lim from the left does not equal the lim from the right.

The limit as x->0 from the right of sqrt(x) is 0. However, limit as x->0 from the left of sqrt(x) is does not exist, and since the limit from the left does NOT equal the limit from the right, then the limit from both sides does not exist.

Conversely, the limit as x->1 from the right of sqrt(x) is 1. And, limit as x->1 from the left of sqrt(x) is 1, and since the limit from the left DOES equal the limit from the right, then the limit from both sides is also 1.(2 votes)

- What Happens if you try to take the limit of a side-ways parabola (like x=y^2+y+1)? Would it also be undefined?(2 votes)
- A side-ways parabola is not a function with respect to x, so we do not study its properties using calculus.(2 votes)

## Video transcript

So we have a function, f of
x, graphed right over here. And then we have a
bunch of statements about the limit of f of x, as
x approaches different values. And what I want to do is figure
out which of these statements are true and which
of these are false. So let's look at
this first statement. Limit of f of x, as x approaches
1 from the positive direction, is equal to 0. So is this true or false? So let's look at it. So we're talking about
as x approaches 1 from the positive direction,
so for values greater than 1. So as x approaches 1 from
the positive direction, what is f of x? Well, when x is, let's say 1
and 1/2, f of x is up here, as x gets closer and closer
to 1, f of x stays right at 1. So as x approaches 1 from
the positive direction, it looks like the limit of
f of x as x approaches 1 from the positive
direction isn't 0. It looks like it is 1. So this is not true. This would be true
if instead of saying from the positive
direction, we said from the negative direction. From the negative direction,
the value of the function really does look like
it is approaching 0. For approaching 1 from the
negative direction, when x is right over
here, this is f of x. When x is right over
here, this is f of x. When x is right over
here, this is f of x. And we see that
the value of f of x seems to get closer
and closer to 0. So this would only
be true if they were approaching from
the negative direction. Next question. Limit of f of x, as x approaches
0 from the negative direction, is the same as limit of
f of x as x approaches 0 from the positive direction. Is this statement true? Well, let's look. Our function, f of
x, as we approach 0 from the negative
direction-- I'm using a new color--
as we approach 0 from the negative direction,
so right over here, this is our value of f of x. Then as we get closer, this
is our value of f of x. As we get even closer, this
is our value of f of x. So it seems from the
negative direction like it is approaching
positive 1. From the positive direction,
when x is greater than 0, let's try it out. So if, say, x is 1/2,
this is our f of x. If x is, let's say,
1/4, this is our f of x. If x is just barely larger
than 0, this is our f of x. So it also seems
to be approaching f of x is equal to 1. So this looks true. They both seem to be
approaching the limit of 1. The limit here is 1. So this is absolutely true. Now let's look at
this statement. The limit of f of
x, as x approaches 0 from the negative
direction, is equal to 1. Well, we've already
thought about that. The limit of f of
x, as x approaches 0 from the negative
direction, we see that we're getting
closer and closer to 1. As x gets closer and
closer to 0, f of x gets closer and closer to 1. So this is also true. Limit of f of x, as x
approaches 0 exists. Well, it definitely exists. We've already established
that it's equal to 1. So that's true. Now the limit of f of x
as x approaches 1 exists, is that true? Well, we already saw that
when we were approaching 1 from the positive
direction, the limit seems to be approaching 1. We get when x is 1
and 1/2, f of x is 1. When x is a little bit
more than 1, it's 1. So it seems like we're getting
closer and closer to 1. So let me write that down. The limit of f of
x, as x approaches 1 from the positive
direction, is equal to 1. And now what's the limit
of f of x as x approaches 1 from the negative direction? Well, here, this is our f of x. Here, this is our f of x. It seems like our f of x is
getting closer and closer to 0, when we approach 1 from
values less than 1. So over here it equals 0. So if the limit from
the right-hand side is a different value than the
limit from the left-hand side, then the limit does not exist. So this is not true. Now finally, the limit of f
of x, as x approaches 1.5, is equal to 1. So right over here. So everything we've been
dealing with so far, we've always looked at
points of discontinuity, or points where maybe the
function isn't quite defined. But here, this is kind
of a plain vanilla point. When x is equal to 1.5,
that's maybe right over here, this is f of 1.5. That right over
there is the point, well, this is the
value f of 1.5. We could say f of, we could see
that f of 1.5 is equal to 1, that this right here is
the point 1.5 comma 1. And if we approach it
from the left-hand side, from values less than it, it's
1, the limit seems to be 1. When we approach from
the right-hand side, the limit seems to be 1. So this is a pretty
straightforward thing. The graph is continuous
right there, and so really, if we just substitute
at that point, or we just look at
the graph, the limit is the value of
the function there. You don't have to have a
function be undefined in order to find a limit there. So it is, indeed, the
case that the limit of f of x, as x approaches
1.5, is equal to 1.