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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

  • squared minus 10.

  • Now, what I want to think about is--

  • 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.

  • Or you could reason through this.

  • 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-- let's just

  • focus on the numerator.

  • 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

  • than any of these other things.

  • Something squaring gets large.

  • But something being raised to the fifth power

  • gets raised that much, much faster.

  • 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

  • other terms.

  • 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.

  • So as x gets very, very, very large,

  • this thing is going to approximate 4x

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

  • Or we could say as x approaches infinity.

  • Now, what could this be simplified to?

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

  • These are going to grow together.

  • So these you can think of them as canceling out.

  • And so you are left with 2/3.

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

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

  • and larger, all of these other terms

  • aren't going to matter that much.

  • And so it's going to approach 2/3.

  • Now, let's look at the graph and see

  • if that actually makes sense.

  • What we're actually saying is that we

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

  • So lets look at the graph.

  • So right here is the graph.

  • Got it from Wolfram Alpha.

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

  • of x seems to be approaching this value that looks right

  • at around 2/3.

  • So it looks like we have a horizontal asymptote

  • right over here.

  • Let me draw that a little bit neater.

  • We have a horizontal asymptote right at 2/3.

  • So let me draw it as neatly as I can.

  • 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

  • and closer to 2/3.

  • And when we just look at the graph here,

  • it seems like the same thing is happening

  • from the bottom direction, when x approaches negative infinity.

  • So we could say the limit of f of x,

  • as x approaches negative infinity, that

  • also looks like it's 2/3.

  • And we can use the exact same logic.

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

  • as it becomes further and further

  • to the left on the number line, the only terms

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

  • and the 6x to the fifth.

  • So this is true for very large x's.

  • It's also true for very negative x's.

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

  • this is also true.

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

  • is going to cancel out.

  • These are the dominant terms.

  • And we're going to get it equaling 2/3.

  • 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,

  • we get 2/3.

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

  • is 2/3.

  • So in general, whenever you do this,

  • you just have to think about what

  • terms are going to dominate the rest?

  • And focus on those.

So we have f of x equaling 4x to the fifth minus 3x squared

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