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U0S1V04 One-sided limits.txt
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#
# File: content-mit-18-01-1x-captions/U0S1V04 One-sided limits.txt
#
# Captions for MITx 18.01.1x module [fAAzAVHVKQk]
#
# This file has 102 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
Welcome back.
We've been thinking about this function.
And in the last video, we took some values
of x that were approaching 1 from the left,
and we made this table.
And we saw that as x approached 1
from the left, f of x approached minus 2.
So remember, these arrows mean approaching.
And this minus sign up here, that
signifies that we're coming at 1 from the left,
or from the negative direction.
This is not the same as negative 1.
x is actually approaching positive 1.
It's just that x is coming from the negative direction.
And as it does that, f of x is approaching negative 2.
What you were supposed to do was the same thing,
just on the other side.
So you should have picked out some values and made a table.
Now, you didn't have to choose these particular values.
But you should have chosen some similar looking ones
and gotten a similar looking table.
And what we see from the table is
that as x approaches 1 from the right,
f of x approaches 2, positive 2, not negative 2.
So we've got something different going on on the right side of 1
versus the left side of 1.
Pretty cool.
Let's see what this looks like on the graph of f.
All of these data points that we have will help us get started.
For instance, on the right here, we've got f of 2 equals 2.24.
So the point (2, 2.24) is on the graph.
And we can do the same thing with these other three points
that are on the right.
And they'll look like this.
Now, it seems reasonable to assume that the graph of f
is going to behave pretty smoothly in between these four
points.
So we can just draw it like this.
And once we've done that, then we can look and see
what happens as x approaches 1 from the right, what's
happening to f of x and what's happening to these y values.
Well, like we said, they're approaching
the level y equals 2.
What happens exactly at x equals 1?
Well, remember that f of 1 isn't defined.
We would have had a 0 denominator, which
means that there is no point on the graph where
the x-coordinate is 1.
So we're going to put this open circle here
to remind us that this point, (1,2)
is not actually on the graph.
But that's OK.
If we're just talking about x approaching 1,
that means we only care about values of x that are near 1,
not equal to 1.
We can do the same thing on the left.
Our table of values gives us these four points.
And if we interpolate and assume the graph
is smooth in between those points, then we've got this.
And as we come in towards 1 from the left with our x values,
then our y values are approaching y equals minus 2.
And again, we'll have this open circle
to remind us that there is no point at x equals 1
on the graph.
OK, let me make some space here.
And we want to give an official name
for this phenomenon of a function's value
approaching something.
We're going to call this a limit.
So on the right here, we're going
to say that the limit of f of x as x approaches 1
from the right is 2.
And the notation for this is as follows.
Limit of f of x as x approaches 1 from the right is 2.
This is often called the right-sided limit
or the right-hand limit at the point x equals 1.
And we have a similar thing on the left.
We can write the limit of f of x as x approaches 1
from the left equals minus 2.
And there we have it.
We've got our left-hand limit, and we've
got our right-hand limit.
We have the notation, we have the meaning,
and we know what it looks like in pictures.
So this is our left-handed limit.
And here's our right-handed limit.
So that's our first function.
Kind of an interesting little function, isn't it?
We have a couple short questions for you.
And then we have a few more functions for you
to play around with left- and right-hand limits.
And maybe those will be even more interesting.
So why don't you go and find out?