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Vipul: In this talk, I'm going to give definitions
of one-sided limits.
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So it is going to be the left hand limit and
the right hand limit, and I'm going to basically
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compare it with the definition of two-sided limit which was in
a previous video. Let's just write this down--left-hand limit.
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Let me first remind you what the definition
of two-sided limit says.
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So here's what it says. It says limit as x approaches
c, f(x) = L
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so f has to be defined on the immediate left and
the immediate right of c.
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It says that this is true if the following
holds so for every epsilon greater than zero
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there exists a delta > 0 such that for all
x which are within delta of c
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either delta on the left of c or within a delta on the
right of c we have that f(x) is within an epsilon
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distance of L.
Okay. Now with the left and right hand limit
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what we are trying to do we are trying to
consider only one-sided approaches on the, on the x
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What will change when we do the left-hand limit,
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what will be different from this definition?
[ANSWER!]
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Rui: We approach c from the left.
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Vipul: We'll approach c from the left so
what part of this definition will change? [ANSWER!]
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Rui: From the fourth line?
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Vipul: You mean this line?
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Rui: Oh for all x within c distance, within delta distance of c
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Vipul: So what will change?
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Rui: We will not have (c, c + delta).
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Vipul: This part won’t be there. We will
just be concerned about whether when x is
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delta close on the left side of c, f(x) is here...
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Will we change this one also to only include the left? [ANSWER!]
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Or this one will remain as it is?
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Rui: I think it will remain.
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Vipul: It will remain as it is because we
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are just saying as x approaches c from the left
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f(x) approaches L.
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We are not claiming that f(x) approaches L
from the left, okay? Let me make a number line picture.
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We will do a full geometric understanding
of the thing later. Right now it's just very [formal].
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So the function is defined on the immediate left
of c, maybe not defined at c. It is defined
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on the immediate left of c.
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We don’t even know if the function
is defined on the right of c and what we are
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saying is that for any epsilon, so any epsilon
around L you can find a delta such that if you restrict
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attention to the interval from c minus delta
to c [i.e., (c- delta, c) in math notation]
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then the f value there is within the epsilon distance of L.
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Now the f value could be epsilon to the left
or the right so we take left hand limit on
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the domain side it doesn’t have to approach
from the left on the other side.
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Let me just write down the full definition. We want to keep this on the side.
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What it says that for every epsilon > 0 there
exists
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by the way, the understanding of the what this definition
really means will come in another video you may have seen before this or after this
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... for all x ... [continuing definition]
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Now we should also change it if we are writing
in this form so how will it read now?
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Rui: For all x ...
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Vipul: So will you put x – c or c – x? [ANSWER!]
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Rui: It will be x – c, oh c – x.
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Vipul: c – x. Because you want c to be bigger
than x. You want x to be on the left of c.
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What would this read, i.e. x is in (c – delta,c).
Okay.
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What do we have? We have the same thing. This part doesn’t change.
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Rui: f(x) is within epsilon distance of L.
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Vipul: Why do I keep saying this thing about the
L approach doesn’t have to be from the left?
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What’s the significance of that? Why is that important?
[ANSWER!]
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Rui: It’s important because we don’t know
whether the function is decreasing or increasing
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at that point.
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Vipul: Yeah, so if your function is actually
increasing than L will also be approached
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from the left, and if it’s decreasing it
will be approached from the right, but sometimes
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it’s neither increasing nor decreasing, but it's still
true it approaches from one side, so that’s a
little complicated but the way
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this comes up is that when you are dealing
with composition of functions, so when you
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are doing one function and then applying another function to that and you have some results
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with one-sided limits.
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Let me just write this down. If you have one-sided
limits and you have composition,
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so you are doing one function and then doing another
you have to be very careful.
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You need to be very careful when you are doing
one-sided limits and composition.
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Why? Because if you have g of f(x) and x approaches
to c from the left, f(x) approaches L but
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not necessarily from the left.
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You then you have another thing which is as
f(x) approaches L from the left, g of that
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approaches something you just need to be careful
that when you compose things the sidedness
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could change each time you compose.
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Rui: Can you write a composition of the function
out?
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Vipul: Not in this video. We will do that
in another video.
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That’s something we will see in a subsequent
video but this is just something to keep in
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mind so when you see that it will ring a bell.
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Let us do... what other side is left? [pun unintended!]
Rui: Right?
Vipul: Right!
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Vipul: By the way, you probably already know
this if you have seen limits intuitively so
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I'm not stressing this too much but left hand
limit is really the limit as you approach
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from the left. You are not moving toward the
left you are moving from the left to the point.
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Right hand limit will be approach from the
right to the point so it is right, moving from
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the right, so the words left and right are
describing where the limit is coming *from*,
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not the direction which it is going to.
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Now you can just tell me what will be the
corresponding thing. To make sense of this
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notion we need f to be defined where? [ANSWER!]
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Rui: On its right.
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Vipul: On the immediate right of c. If it
is not defined on the immediate right it doesn’t
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even make sense to ask this question what
the right hand limit is.
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How will that be defined?
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Rui: For every epsilon greater than zero
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Vipul: The epsilon is the interval on which
you are trying to trap the function value.
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Rui: There exists epsilon
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Vipul: No, delta
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Rui: delta> 0 such that for all x
with x – c > 0
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Vipul: The general one is for all x with 0<|x-c|<delta
because you want to capture both the intervals.
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In this one, the left hand limit one, we just
captured the left side interval.
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Now in the right one we just want to capture
the right side interval, so as you said 0< x- c < delta.
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In the picture, the function is defined, say c
to c + t and you are really saying you can
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find delta if x is in here [between c and c + delta] which
actually... this is not including c, it is all the points
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in the immediate right of c. We have? [ANSWER!]
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Rui: The absolute value of f(x) – L is less
than epsilon.
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Vipul: So f(x) is? Are we here? We have everything?
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Rui: Yes.
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Vipul: We have both of these here? So do you
see what’s the main difference between these
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two and the actual [two-sided limit] definition?
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For every epsilon there exists delta... the
first second and fourth line remain the same.
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It is this line where you are specifying where
the x are that’s different.
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In the two-sided thing the x could be either place.
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For the left hand limit the x,
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you just want x here [in (c - delta, c)] and
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for the right hand limit you just want x in (c,c + delta).
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Okay? [END!]