- [Instructor] What we're going to do in this video
is use the online graphing calculator Desmos,
and explore the relationship between vertical
and horizontal asymptotes, and think about
how they relate to what we know
about limits.
So let's first graph two over x minus one,
so let me get that one graphed,
and so you can immediately see that something interesting
happens at x is equal to one.
If you were to just substitute x equals one
into this expression, you're going to get
two over zero, and whenever you get a non-zero thing,
over zero, that's a good sign that you might
be dealing with a vertical asymptote.
In fact we can draw that vertical asymptote
right over here at x equals one.
But let's think about how that relates to limits.
What if we were to explore the limit as x approaches one
of f of x is equal to two over x minus one,
and we could think about it from the left
and from the right, so if we approach one
from the left, let me zoom in a little bit
over here, so we can see as we approach from the left
when x is equal to zero, the f of x
would be equal to negative two,
when x is equal to point five, f of x
is equal to negative of four, and then it just gets
more and more negative the closer we get
to one from the left.
I could really, so I'm not even that close yet
if I get to let's say 0.91, I'm still nine hundredths
less than one, I'm at negative 22.222, already.
And so the limit as we approach one from the left
is unbounded, some people would say
it goes to negative infinity, but it's really
an undefined limit, it is unbounded
in the negative direction.
And likewise, as we approach from the right,
we get unbounded in the positive infinity direction
and technically we would say that that limit
does not exist.
And this would be the case when we're dealing
with a vertical asymptote like we see over here.
Now let's compare that to a horizontal asymptote
where it turns out that the limit
actually can exist.
So let me delete these or just erase them for now,
and so let's look at this function
which is a pretty neat function, I made it up
right before this video started
but it's kind of cool looking, but let's think
about the behavior as x approaches infinity.
So as x approaches infinity, it looks like our y value
or the value of the expression, if we said y
is equal to that expression, it looks like
it's getting closer and closer and closer to three.
And so we could say that we have a horizontal asymptote
at y is equal to three, and we could also
and there's a more rigorous way of defining it,
say that our limit as x approaches infinity
is equal of the expression or of the function,
is equal to three.
Notice my mouse is covering it a little bit
as we get larger and larger, we're getting
closer and closer to three,
in fact we're getting so close now, well here
you can see we're getting closer and closer
and closer to three.
And you could also think about what happens
as x approaches negative infinity and here
you're getting closer and closer and closer
to three from below.
Now one thing that's interesting about horizontal
asymptotes is you might see that the function
actually can cross a horizontal asymptote.
It's crossing this horizontal asymptote
in this area in between and even as we approach infinity
or negative infinity, you can oscillate
around that horizontal asymptote.
Let me set this up, let me multiply this times sine of x.
And so there you have it, we are now oscillating
around the horizontal asymptote,
and once again this limit can exist
even though we keep crossing the horizontal asymptote,
we're getting closer and closer and closer to it
the larger x gets.
And that's actually the key difference between
a horizontal and a vertical asymptote.
Vertical asymptotes if you're dealing with a function,
you're not going to cross it, while with a horizontal
asymptote, you could, and you are just getting
closer and closer and closer to it
as x goes to positive infinity or as x
goes to negative infinity.