Name: Limits at infinity of quotients (Part 2) | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T14:04:51.000Z
Duration: 4 min 33 s
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Let's do a few more examples of finding the limit of functions

as x approaches infinity or negative infinity.

9x to the seventh minus 17x to the sixth,

All of that over 3x to the seventh plus 1,000x

So what's going to happen as x approaches infinity?

And the key here, like we've seen in other examples,

is just to realize which terms will dominate.

out of these three terms, the 9x to the seventh

is going to grow much faster than any of these other terms.

So this is the dominating term in the numerator.

And in the denominator, 3x to the seventh

is going to grow much faster than an x to the fifth term,

and definitely much faster than a log base 2 term.

So at infinity, as we get closer and closer to infinity,

this function is going to be roughly equal to 9x

as we get larger and larger as we get closer and closer

going to get closer and closer each other.

Well, we can just cancel out the x to the seventh.

So that is our limit, as x approaches infinity,

Now let's do the same with this function over here.

Which terms dominate as the absolute value of x

Well, in the numerator, it's the 3x to the third term.

In the denominator it's the 6x to the fourth term.

So this is going to be the same thing as the limit of 3x

to the third over 6x to the fourth, as x approaches

going to be equal to the limit as x approaches

Well, if the denominator, even though it's

becoming a larger and larger and larger negative number,

it becomes 1 over a very, very large negative number.

Which is going to get us pretty darn close to 0.

Just as 1 over x, as x approaches negative infinity,

So this right over here, the horizontal asymptote

And I encourage you to graph it, or try it out with numbers

The key realization here is to simplify the problem

Well, once again, what are the dominating terms?

In the numerator, it's 4x to the fourth, and in the denominator

So this is going to be the same thing as the limit,

as x approaches infinity, of 4x to the fourth over 250x

Which is going to be the same thing as the limit of-- let's

going to be the same thing as-- well we could divide two hundred

It's going to be the limit of 4 over 250.

x to the fourth divided by x to the third is just x.

Or we could even say this is going to be 4/250 times

the limit, as x approaches infinity of x.

What's the limit of x as x approaches infinity?

Well, it's just going to keep growing forever.

So this is just going to be, this right over here

Infinity times some number right over here

So the limit as x approaches infinity of all of this,

And a kind of obvious way of seeing that, right,

from the get go, is to realize that the numerator has

While the highest degree term in the denominator

So the numerator is going to grow far faster

So if the numerator is growing far faster