# Thread: Help with finding the asymptote on a rational equation

1. ## Help with finding the asymptote on a rational equation

Hi, thanks in advance for any attempts to help me with this question I've been stuck on for a while now. The function is as follows;
f(x) = (vKx)/(K+(v-1)x)
Where x is the population in a given generation, f(x) is the population in the next, and v>1 and K>0 are constants. Because x represents a population, the domain is restricted to x≥0. As aforementioned I'm having difficulty finding the horizontal asymptote.
I assume that one should start by setting a limit as x approaches infinity, but I'm unsure where to take it from there.

2. ## Re: Help with finding the asymptote on a rational equation

$f(x)=\dfrac{v K x}{(K+(v-1)x)}$

$x \to \infty \Rightarrow (K+(v-1)x) \approx (v-1)x$

so

$x \to \infty \Rightarrow f(x) \approx \dfrac{v K x}{(v-1)x} = \dfrac{v}{v-1}K$

3. ## Re: Help with finding the asymptote on a rational equation

Thanks! That certainly seems to work. May I ask why/how you "removed" the K in the denominator?

4. ## Re: Help with finding the asymptote on a rational equation

Originally Posted by Daimpas
Thanks! That certainly seems to work. May I ask why/how you "removed" the K in the denominator?
The usual way to show this is to divide top and bottom by $x$

\begin{align*} \displaystyle &\lim_{x\to \infty} \dfrac{v K x}{K+(v-1)x} = \\ \\ &\lim_{x\to \infty} \dfrac{v K }{\frac K x +(v-1)} = \\ \\ &\dfrac{v K}{v-1} \end{align*}