Horizontal and vertical asymptotes of function

Voiceover:Right over here,

I have the graph of f of x,

and what I want to think
about in this video

is whether we could
have sketched this graph

just by looking at the
definition of our function,

which is defined as a rational expression.

We have 2x plus 10 over 5x minus 15.

There is a couple of ways to do this.

First, you might just want to pick out

any numbers that are
really easy to calculate.

For example, what happens
when X is equal to 0?

We could say f of 0 is
going to be equal to,

well, all the x term is going to be 0,

so you're going to be left
with 10 over negative 15,

which is negative 10/15,

which is negative 2/3.

You can plot that one.

x equals 0.

f of x or y equals f of x is negative 2/3,

and we see that point,

let me do that in a darker color,

you see that point right over there,

so we could have plotted that point.

We could also say, "When
does this function equal 0?"

Well, the function is equal to 0 when ...

The only way to get
the function equal to 0

is if you get this numerator equal to 0,

so you could try to solve
2x plus 10 is equal to 0.

That's going to happen when
2x is equal to negative 10.

I just subtracted 10 from both sides.

If I divide both sides by 2,

that's going to happen when
x is equal to negative 5.

You see that, you see
this right over here.

When x is equal to negative 5,

the function intersects the x-axis.

That's just two points,

but that still doesn't give us enough

to really form this
interesting shape over here.

You could think about what other functions

have this type of shape.

Now what I want to think
about is the behavior

of the function at different points.

First, I want to think
about when this function

is undefined and what type
of behavior we might expect

for that function when it's undefined.

This function is going to be undefined.

The only way I can think
of to make this undefined

is if I make the denominator equal to 0.

We don't know what it
means to divide by 0.

That is undefined.

The function is going to be undefined

when 5x,

let me do this in blue,

when 5x minus 15 is equal to 0,

or adding 15 to both sides,

when 5x is equal to 15,

or dividing both sides by 5,

when x is equal to 3,

f is undefined.

Now, there is a couple of ways

for a function to be undefined at a point.

You could have something like this.

Let me draw some axes right over here.

Let's say that this is 3.

You could have your function,
it could look like this.

It could be defined.

It might approach something
but just not be defined

right at 3 and then just
keep on going like that,

or the other possibility is it might have

a vertical asymptote there.

If it has a vertical asymptote,

it's going to look something like this.

It might be approach, it
might just pop up to infinity,

and it might pop down from
infinity on this side,

or it might go from negative
infinity right over here.

That's what a vertical
asymptote would look like,

that as we approach from the left,

the graph is approaching a vertical,

but it never quite gets to x equals 3,

I guess one way we could say it,

or the function is not
defined at x equals 3.

As you approach from the
right, the same thing,

the function just, in
this case, drops down.

It almost becomes vertical.

It's approaching negative infinity

as x approaches 3 from
the positive direction.

So how would we know?

Obviously, when you look at here,

when we know the graph ahead of time,

and if you say, "OK, this is easy."

This is x equals 10.

Let's see how many.

This is 1, 2, 3, 4, 5,

so each of these are 2,

so x equals 3 is right over here.

When you look at the graph,

if you had the graph in front of you,

you would see,

"Oh, look, this is indeed
a vertical asymptote."

Just looking at the graph,

you see that you have a vertical
asymptote at x equals 3,

and let me write that down,

vertical asymptote,

vertical asymptote,

asymptote at x equals 3,

at x equals 3,

but how would you have known that?

How would you have known that

if you didn't have the graph here,

if you just had this?

We know it's not defined at 3,

but how do we know it's
not a point discontinuity

and not instead a vertical asymptote?

There's a couple of ways to do it.

One way is you could try values near 3

and see what happens.

For example, you could
get your calculator out,

and you could try, let's say, 3.01.

If you say 2 times 3.01 plus 10,

actually, oh, that's the numerator,

and then I'm going to divide that,

the numerator, by 5 times 3.01 minus 15.

It gets us a fairly large number.

It's exploding on us.

If we got even closer, so
if we did 2 times 3.001,

plus 10 divided by 5 times 3.001,

now I'm trying x is
equal to 3.001 minus 15,

we see we get even a larger number.

As x gets closer and closer to 3,

f of x seems to be exploding.

It seems to be approaching
positive infinity.

That's one way to say,
OK, this looks like,

at least from this side,
we are approaching,

we are approaching positive infinity,

so we would have been able
to draw something like that,

and then you could have
tried values below.

You could have tried values below,

so you could have said ...

Actually, let me just
put the last entry here,

and let me just change
the 3.001s to 2.999,

2.999,

and whoops, let me go over here.

We have 2.999, and we get ...

We're going really negative now.

We're approaching negative infinity.

If you just tried that out,

that would give you a
pretty good indication

that the graph is looking
something like that

right over here,

which also seems to match
connecting these two points

that we've already thought about.

But now let's see what's going on

as x approaches really large values

or really positive values
or really negative values.

It looks like there is a
horizontal asymptote here.

Just looking at the graph,

it seems like there is some value

that as x approaches really large values,

really positive values,

f of x is going to be
approaching that value,

that asymptote from above.

As x becomes really negative,

it looks like f of x is
approaching that from below.

But how would we be
able to figure that out

just by looking at this?

One thought experiment is just to say

what happens to f of x
as x approaches infinity?

Let me write that down.

As x approaches infinity,

then f of x is going to approach what?

As x approaches larger and larger values,

the positive 10 and the negative 15

start to matter a lot less.

At the highest degree term,

the numerator and denominator
start to dominate.

We could say as x approaches infinity,

f of x is getting closer
and closer to 2x over 5x,

which is 2/5,

which is equal to 2/5.

You could say f of x is approaching 2/5.

If you really want to see that
a little bit more concretely,

let's imagine different values for x,

as x gets larger and larger and larger.

If we have,

so x, f of x.

If x is 1, then f of x is
just going to be 2 plus 10

over 5 minus 15.

Here, the 10 and the
subtracting the 15 matter a lot.

But if x were 1,000,

then f of x would be 2,000
plus 10 over 5,000 minus 15.

Now, the 2,000 and 5,000 are
really setting the agenda.

Then if x were, let's say, 1 million,

1 million,

and I just use blue for more contrast,

then f of x would be 2 million,

2 million plus 10,

let me move over to
the right a little bit,

2 million plus 10 over 5 million minus 15.

Here, the 10 and the 15
are almost inconsequential.

You can imagine if x were a
billion or a trillion or a google,

then the 10 and the
negative 15 start to matter

a lot and lot less.

As x approaches infinity,
these matter less.

The highest degree terms matter,

so f of x is going to approach 2x over 5x,

which is 2/5.

So f of x is approaching 2/5,

and that's what this line looks like.

2/5 is same thing as 0.4, so f of x,

and we see that in the graph.

f of x is approaching that,

but it's not quite getting close to it.

It's not quite getting there.

It's getting closer and closer
to it as x goes to infinity,

but it's not quite getting there

because you're always going to have

that plus 10 and that minus 15 there,

so you're never going
to be exactly at 2/5.

The same thing is happening

as x gets more and more and more negative.

You can make all of these negative values.

If this were negative 1,

that would be negative 2, negative 5.

If this were negative 1,000,

it would be negative
2,000 over negative 5,000.

If it's negative 1 million,

it would be negative 2 million
over negative 5 million.

But you see, even in this case,

f of x is approaching 2x over
5x, which is approaching 2/5,

or you could say it's approaching
negative 2 over negative 5

which is still 2/5, and you
see that right over here.

We would say that this function
has a horizontal asymptote,

horizontal asymptote,

at y equals a horizontal
line right over here,

y is equal to 2/5.

Hopefully, this graph here
is helping us appreciate

what these vertical and
horizontal asymptotes

actually are.

But if we didn't have the graph,

we could have said, OK, we're undefined

at x equals 3.

We could test some values around it,

and so we could say, OK, look,

it does look like we're
approaching negative infinity

as x approaches negative 3 from the left.

It looks like we're
approaching positive infinity

as x approaches negative 3 from the right,

so we could draw that
blue point right there.

We could graph these two points,

when does f equal 0,

What happens to f when x equals 0,

and then we could think about the behavior

as x approaches infinity
or negative infinity,

as x approaches infinity
or negative infinity,

and draw this horizontal asymptote.

Between all of those,

that would have been a pretty good way

to be able to sketch this actual graph.