Name: Limits at infinity of quotients (Part 1) | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T14:04:51.000Z
Duration: 4 min 7 s
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So we have f of x equaling 4x to the fifth minus 3x squared

plus 3, all of that over 6x to the fifth minus 100x

what is the limit of f of x, as x approaches infinity?

And there are several ways that you could do this.

You could actually try to plug in larger and larger numbers

for x and see if it seems to be approaching some value.

And when I talk about reasoning through this,

it's to think about the behavior of this numerator

and denominator as x gets very, very, very large.

And when I'm talking about that, what I'm saying is,

As x gets very, very large, this term right

over here in the numerator-- 4x to the fifth--

is going to become a much, much more significant

But something being raised to the fifth power

Similarly, in the denominator, this term right over here-,,

the highest degree term-- 6x to the fifth--

is going to grow much, much, much faster than any of these

Even though this has 100 as a coefficient or a negative 100

as a coefficient, when you take something to the fifth power,

it's going to grow so much faster than x squared.

to the fifth over 6x to the fifth for a very large, large x

Or we could say as x approaches infinity.

Well, you have x to the fifth divided by x to the fifth.

So these you can think of them as canceling out.

So what you could say is-- the limit of f

of x, as x approaches infinity, as x gets larger and larger

have a horizontal asymptote at y is equal to 2/3.

And we see, indeed, as x gets larger and larger and larger, f

of x seems to be approaching this value that looks right

So it looks like we have a horizontal asymptote

We have a horizontal asymptote right at 2/3.

So this right over here is y is equal to 2/3.

The limit as x gets really, really large,

as it approaches infinity, y is getting closer and closer

it seems like the same thing is happening

from the bottom direction, when x approaches negative infinity.

When x becomes a very, very, very negative number,

to the left on the number line, the only terms

that are going to matter are going to be the 4x to the fifth

So we could also say, as x approaches negative infinity,

And then, the x to the fifth over the x to the fifth

And once again, you see that in the graph here.

We have a horizontal asymptote at y is equal to 2/3.

We take the limit of f of x as x approaches infinity,

And the limit of f of x as x approaches negative infinity