Undergrad Asymptotes of Rational Functions....

[fog37]

TL;DR

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!

 

[mfb]

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).

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.

 

[fog37]

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!

 

[mfb]

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