U0S1V07 Possible limit behaviors.txt

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You've now seen a variety of limits.
In our last video, we looked at a function f
whose limit as x approached a point a from the right
was equal to 2.
And so it looked something like this.
And the limit from the left as x approached a was minus 2.
So it looked something like this.
So we know that the right- and the left-hand limits at a point
don't have to agree, but they could agree.
One of the examples that you looked at was like that.
That was the function that we called g.
There we had the left-hand limit equal to 1/2.
So the graph would look like this.
And the right-hand limit was also equal to 1/2.
So the graph would look something like that.
Now, of course, for the purposes of limits,
it wouldn't have mattered what g of a itself was.
g of a could of been down here or it
could've been equal to the value of the limit in which case
we would have had a dot up here.
For the actual function, I think g of a didn't exist at all,
but that's OK.
That's the great thing about limits,
which is that if you have a formula or a function that
doesn't exist or doesn't work when you try
to plug in some value a, you can still say something
about how it behaves near a by taking limits.
But sometimes the limits themselves don't exist.
So you saw another function h where
as x approached a point from the left,
the limit was perfectly fine.
And I think, it was 0, but coming in from the right
the value of the function just kept
getting bigger, and bigger, and bigger without bound.
So its graph would have had this sort of vertical asymptote
as we come in from that side towards a.
So in this case, when h doesn't approach any particular value
coming in from the side, we say that the limit does not exist.
And we'll always write DNE for does not exist.
So that's one way in which a one-sided limit might not
exist.
The function just kind of blows up towards infinity as we
come in from one side.
Of course, we could have had h going the other way.
It could have blown up towards minus infinity
and that would also be a limit that doesn't exist.
But in the last function that we gave you, which we called j,
was something that was even more bizarre.
So there we had a function and it didn't get very big,
either positive nor negative.
In fact, I don't know if you noticed,
but it was always between the values y equals 1
and y equals minus 1, but it never settled down.
As x was coming in towards the value,
hopefully, you saw that j of x just
kept bouncing back and forth, going up and down, faster
and faster, until it just kind of goes haywire as x comes in
really, really close to a.
So, yeah, really weird.
We'll try not to give you too many functions like this,
but you should be aware that stuff like this can happen.
And when it does we'll also say that the limit from the right
does not exist.
So we have some really quick review questions for you.
And then when we come back, we'll take our right-hand limit
and we'll take our left-hand limit
and we'll put them together and we'll have an overall limit.
So stay tuned for that.