Asymptotes of Rational Functions....

In summary, rational functions with a numerator degree greater than the denominator degree by one unit will have one slant asymptote and no horizontal asymptotes, but possibly several vertical asymptotes. The number of slant asymptotes can increase if the numerator degree is greater than the denominator degree by two or more units. A curve can also cross its asymptotes, as long as it remains around the asymptotic line. Curved asymptotes are possible, as seen in the example $$\frac {x-x^2+x^4}{x^2-1}$$ and can also intersect their asymptotes, as shown in the function (-0.1*x^3+1x^2-3.5x+4)/x^2
  • #1
fog37
1,568
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TL;DR Summary
asymptotes of rational functions when numerator degree> denominator degree
Hello,
I know that functions can have or not asymptotes. Polynomials have none.

In the case of a rational functions, if the numerator degree > denominator degree by one unit, the rational function will have a) one slant asymptote and b) NO horizontal asymptotes, c) possibly several vertical asymptotes.
  • How many slant asymptotes can a rational function have when numerator degree > denominator degree by one unit? Always just a single one?
  • What happens if numerator degree >denominator degree by 2 or even more units? Does the rational function still have a single slant asymptote, none or more than one? For example, ##\frac {4x^5+2}{3x+2}##
Finally, I believe that a curve can even cross its asymptotes, either nearby or even far away (far up, far down, far left or far right), as long as the curve "remains around" the asymptotic line. For example, imagine a slant asymptote and the curve oscillating, and crossing it, but the curve navigates along the asymptote itself...

Thanks!
 
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  • #2
Divide by the highest power of x of the denominator and it should be clearer what happens for large x (where 1/x becomes negligible).
fog37 said:
Finally, I believe that a curve can even cross its asymptotes, either nearby or even far away (far up, far down, far left or far right), as long as the curve "remains around" the asymptotic line. For example, imagine a slant asymptote and the curve oscillating, and crossing it, but the curve navigates along the asymptote itself...
For general functions yes, but a polynomial won't oscillate.
 
  • #3
Thank you. I did. When we do polynomial division, we get $$Q(x)+R(x)$$

the quotient ##Q(x)## is not a straight line. We neglect the remainder #R(x)## since it becomes extremely small. So far we have assumed that asymptotes are straight lines.

But what about curved asymptotes? I think they exist with no problem. I found this interesting example online: $$\frac {x-x^2+x^4}{x^2-1}$$ which appears to have a curved asymptote...

Thanks!
 
  • #4
Sure, you get ##f(x) \approx x^2## for large |x|.

While you can't get oscillations, here is a function that intersects its asymptote: (-0.1*x^3+1x^2-3.5x+4)/x^2
 

Related to Asymptotes of Rational Functions....

What are asymptotes of rational functions?

Asymptotes of rational functions are lines that a graph approaches but never crosses. They can be vertical, horizontal, or slant (oblique).

How do you find the vertical asymptotes of a rational function?

To find the vertical asymptotes, set the denominator of the rational function equal to zero and solve for the variable. The resulting values are the x-coordinates of the vertical asymptotes.

What is the significance of horizontal asymptotes in rational functions?

Horizontal asymptotes represent the behavior of the function as x approaches positive or negative infinity. They can indicate the overall trend of the graph and help in understanding the end behavior of the function.

Can a rational function have more than one asymptote?

Yes, a rational function can have multiple asymptotes. It can have both vertical and horizontal asymptotes, and in some cases, it can also have slant asymptotes.

How do you determine the slant asymptote of a rational function?

To find the slant asymptote, divide the numerator by the denominator using long division. The resulting quotient is the equation of the slant asymptote.

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