Oblique Asymptotes: What happens to the Remainder?

  • Thread starter LaMantequilla
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In summary, when trying to find the oblique asymptote of a function with a non-linear denominator, we use polynomial long division to get the quotient and remainder. The remainder is disregarded because, as x approaches infinity or negative infinity, it goes to 0 and does not affect the asymptote.
  • #1
Let's say I'm trying to find the oblique asymptote of the function:

f(x)=
-3x2 + 2
x-1

Forgive my poor formatting.

So because the denominator isn't linear, we do polynomial long division of the function and ultimately get -3x - 3 as our quotient, with a remainder of -1. For the sake of the oblique asymptote, we disregard the remainder. But why? I haven't found a website that explains it; they all simply say to ignore the remainder. It bothers me that we just forget about it, regardless of what it is.
Can you explain to me why we disregard the remainder, and where it "goes?"
 
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  • #2
An "asymptote" is a line that a curve approaches as x goes to, in this case, negative infinity and infinity. Yes, long division gives a quotient of -3x- 3 with a remaider of -1. That means that
[tex]\frac{-3x^2+ 2}{x- 1}= -3x- 3- \frac{1}{x- 1}[/tex]
As x goes to either infinity or negative infinity, that last fraction goes to 0.
 
  • #3
That makes perfect sense! Thank you so much!
 

1. What is an oblique asymptote?

An oblique asymptote is a type of asymptote that is a straight line that the graph of a function approaches but never touches. It is also known as a slant asymptote.

2. How is an oblique asymptote different from a vertical asymptote?

While a vertical asymptote represents a value that the function cannot approach, an oblique asymptote represents a value that the function approaches as the input gets larger or smaller.

3. How do you find the equation of an oblique asymptote?

To find the equation of an oblique asymptote, you can use the long division method. Divide the numerator by the denominator of the function and the resulting quotient will be the equation of the oblique asymptote.

4. What happens to the remainder when finding an oblique asymptote?

The remainder is the difference between the original function and the oblique asymptote. When finding an oblique asymptote, the remainder becomes 0, indicating that the function is getting closer and closer to the asymptote.

5. Can a function have more than one oblique asymptote?

Yes, a function can have multiple oblique asymptotes depending on the behavior of the function at infinity. It is also possible for a function to have both vertical and oblique asymptotes.

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