I'm going to do so much to show you **how to find vertical and horizontal asymptotes** of a given
function. So the first thing we want to do is when we ‘refighting a vertical
asymptote our vertical. Asymptote is going to tell us pretty much
you know what values of our function. Are not going to cross on the x-axis so
what's very similar to is actually finding that domain. And if you remember
when you have a rational function to find the domain you determine. What values
make it zero on the bottom and whatever values make it zero are not going to be
a part of your domain. And that's the same thing with the vertical asymptote.
Because if you think of a vertical asymptote it's gonna be a number it's a
vertical line. That it's a value of x that your graph does not have does it is
not defined for so to determine. This I'm going to set each one of my values so
what I want by my vertical very flat. So what I'm going to do is I’m going to
set the bottom equal to zero for each problem. And this is just very similar to
like finding the domain and the reason.

Why is remember whatever values make it 0 that's not part of
my domain and the sense is not part of my domain. It's not an x
value of my function or my function is not defined for that value. Therefore
it’s called what we call an asymptote and meaning that asymptote is when you
look at a graph–. The graph is never going to have a value at that point and
we’ll give it to him. When we talk about graphs if you guys can see what I'm
talking about so here. **How to find
vertical asymptotes. **I'm just going to
add 3xsquared equals 3 square root x equals plus or minus 3. I'm sorry square
root of3 right so therefore my vertical asymptote for this problem. That means
that X values are x equals plus or minus the square root of 3. So we're getting
to grab them later here to find the vertical asymptotes I started equal to 0.
And here since I have a trinomial I can't do adding to the go side. When I have a
trinomial I'm going to looking for a factory and I see that this can be easily
factored in. Into X plus6 and X plus, 1 equals 0 therefore my asymptotes are exposed negative 6. And equal’s negative 1 I'll just try to make a little bit quicker
remember. Just set each one of those equally to zero and then you solve right
here this was very easy. I already know that x equals do with x is 0 is my
vertical asymptote. So the next one is we need to look at our horizontal turn
how to recompense for something else. So what do you want to play your
horizontal asymptote what do you do is we're going to look at there are two
things. We need to look at our leading coefficient so we need to compare our
beating.

Leading coefficients and degrees so you guys look at each
one of these properties. We need to look at the leading coefficient and the
degree of our leading term. **How to find
vertical asymptotes. **So here of each
polynomial so at the top and on the bottom. So differently conversion entries
for your polynomial up top and bottom. Of the bottom here we have a polynomial
but it’s a constant polynomial now. A lot of you might say well there's no X
right there right well I can put X to the zero power. Because any number
erase anything raised to the zero power equals one right what times 3 would be
3. So then the next thing I do is I want to compare my degrees alright and when
my degree up top is less. Then the bottom so when 0 is less than 2 the top is
less than the bottom right of your degrees. My horizontal asymptote is going to
equal 0 and I'm saying why because you know this is your y-axis. And that's
yours-axis so my Y values can equal 0 all right 0 1 so whenever your top
degree. Is less than your bottom degree your leading coefficient other than
that there why is equal to zero?

Okay next separate this map that was one and that was three
whatever. You have at top is mark is smaller than your bottom degree it’s always
like. **How to find vertical asymptotes. **This you're having all this so it doesn't
matter if it's five and fifty it's less than zero. Is less than so life is
really now when you have that people okay what you do is you take the number in
front? Which there's no number in front of your right so we rate one you take
the number in front. And you define them so whenever they’re equal so when you
say like the top is equal to the bottom.

What we do is we take our coefficients and we divide them so
we say y equals 1. Over 1 which is 1 in this case so whenever they're equal so
if this was like. This is like a 3 force than if you three force right so it
just always divide the coefficients and then now. **How to find vertical asymptotes. **We
have we can put a 1 there whenever we have the top that is larger than the bottom.
So you can say the top is larger than the bottom we there is no horizontal
asymptote no horizontal. And we will that means there’s going to be a slant
which we’ll show in another video. But if you guys can just remember whenever
the top degree is smaller than the bottom degree. Of your leading term then why
your horse out last food y equals zero. When they're equal you take the two
coefficients and you divide them and here. What about that one is one and here
whenever it's larger than the bottom. There are no words on trestles and that's a
horizontal and vertical essence.

### Easy Ways To Find The Horizontal Asymptote

Now! I'm going to tell you super **easy ways to find the horizontal asymptote **for any rational
expression that you're given. So basically, there are 3 cases and if you know
those 3, then you're set. So to do this you're going to need to know what the
degree of a polynomial is. So for instance, if your polynomial is: x^2 + 3x +
4The degree is the highest power that appears in your polynomial. So in this
case x^2 is the highest power of x and the degree is 2. The degree is 2, so
it's that PowerBook, so this is a trick to know the horizontal asymptote. It’s
a rule that you can use. So the first case: What if your degree on top is less
than your degree on the bottom? So in this case, the degree on top is 1 for the
x. and 2 for the x^2. If your degree on top is less than the degree on the
bottom.

Then your horizontal asymptote HA will always be y = 0. Which
is the x-axis. So that would be the answer. Whenever the degree on top is less
than the degree in the denominator. Your horizontal asymptote is y = 0. Second
case: If your degree on top is the same as your degree on the bottom. **Easy ways to find the horizontal asymptote.
**In this example that's x^2 on top, x^2 on the bottom so the degree is 2 on
top and 2 on the bottom. Then your horizontal asymptote you will get by
dividing the leading term on top by the leading term on the bottom. So x^2
divided by 3x^2and simplify and this leaves you with 1 over 3, since that was just
a 1x^2. So your horizontal asymptote is y = 1/3So if the degrees are equal on
top and bottom. You’re going to end up with a horizontal asymptote that is: y =
a number. OK, a third case: If your degree on top is larger than your degree on
the bottom you don't have a horizontal asymptote.

So your answer is actually just 'none’. For the horizontal
asymptote specifically, you would write 'none’. Now, in reality, you do have an
asymptote in this case. But it's something called an 'oblique asymptote ‘or a
'slant asymptote ‘and you would actually get it by using long division with
polynomials. You would divide this polynomial by that polynomial and you would
end up with another x expression using long division. **Easy ways to find the horizontal asymptote. **And that would be your
slant asymptote or your oblique asymptote. But you don't need to worry about
that because you’ve been asked about the horizontal asymptote and the answer is
it doesn't have one. So the answer's none.

So there are the 3 cases: It’s going to be one of these 3
cases absolutely. Either Lower degree on top than on bottom and then the
asymptote is: y = 0 Equal degrees and the asymptote is some number. Higher
degree on top than on bottom like a top-heavy rational expression and you has
no horizontal asymptote. So 'none’. So those 3 cases are all you need to
remember. **Easy ways to find the
horizontal asymptote. **That's it! So I think what trips people up is that
they think they need to use some formal definition of the limit to find the
horizontal asymptote. You don’t. For a rational expression, meaning numerator
over denominator for a rational function you really just need to remember these
3 rules. Is the top highest-power, larger than the bottom highest-power? Is it
equal? Is it less than? Those 3 cases. So don't mess with limits! Just remember
the rule. So I know that was a lot of fun.

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