If |x| is large, what is f(x) approximately?

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In summary, when |x| is large, the function f(x)=(x5-x4+x3+x)/(x3-1) is approximately equal to x2-x+1, even though both functions approach infinity. This is because the difference between them becomes smaller and smaller as x gets larger.
  • #1
dumbQuestion
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Homework Statement



If |x| is large, then f(x)=(x5-x4+x3+x)/(x3-1) is approximately what?

Homework Equations



Just use long division


The Attempt at a Solution



Well, I just started out dividing the polys, and I ended up with f(x)=x2-x+1 + (x2+1)/(x3-1)


I thought, well, if x is very large, then the fraction at the end there will begin disapearing and tend towards 0. The solution in the book agreed, but I'm confused about something. The solution in the book says "as the limit as x --> infinity, (x2+1)/(x3-1) = 0, so f(x) ≈x2-x+1 But this doesn't make sense because as x tends to infinity, x2-x+1 will blow up towards infinity. I guess I am just not wrapping my mind around how f(x) is approximately x2-x+1 for |x| "very large". Is it just because the fraction goes to 0 faster than x2-x+1 goes to infinity?
 
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  • #2
Essentially f(x) is infinity +0 to begin, because |x| is large. Work back from that assumption and it seems clear since the limit is infinity
 
  • #3
Well the book was being a bit sloppy - as x gets very large f(x) approaches infinity as the function y=x2-x+1.

It means:
If you plotted the whole f(x), and the quadratic y as a dotted line, you'll find that f(x) gets closer and closer to following the dotted line. That is how f(x) approaches infinity.
 
  • #4
The graphs of f(x) and x2-x+1 will become very close together as x gets large, because the difference between them goes to zero. It doesn't matter that they are both going to infinity. They will still be close together as they do.
 
  • #5
asymptote
 
Last edited:

1. What does the absolute value of x being large tell us about f(x)?

The absolute value of x being large indicates that the value of x is far from zero, and therefore, f(x) is significantly affected by the value of x.

2. How does the function f(x) behave when |x| is large?

When |x| is large, f(x) tends to increase or decrease at a faster rate compared to when |x| is small. This is because the absolute value of x has a greater impact on the function when it is large.

3. Can we use the value of f(x) to approximate the value of x when |x| is large?

No, we cannot use the value of f(x) to approximate the value of x when |x| is large. The absolute value of x being large does not provide enough information to accurately determine the value of x.

4. How does the approximation of f(x) change as |x| becomes larger?

The approximation of f(x) becomes less accurate as |x| becomes larger. This is because the function becomes more sensitive to changes in the value of x as it becomes larger.

5. Is there a specific range of values for x in which the approximation of f(x) is most accurate?

Yes, typically the approximation of f(x) is most accurate when |x| is relatively small. This is because the function is less sensitive to changes in the value of x when it is smaller.

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