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Thread: Help with finding the asymptote on a rational equation

  1. #1
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    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.
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    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

    $x \to \infty \Rightarrow f(x) \approx \dfrac{v K x}{(v-1)x} = \dfrac{v}{v-1}K$
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    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?
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    Re: Help with finding the asymptote on a rational equation

    Quote Originally Posted by Daimpas View Post
    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*}$
    Thanks from Archie
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