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.