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Ive been stuck on this quest from textbook and I can't find an answer for it..

Question is, what may the equation of a rational function with below conditions look like? Explain how you got the equation.

$\displaystyle f(1)=0$

$\displaystyle \lim_{x \to 0} \; f(x)= - \infty$

$\displaystyle \lim_{x\to2^+} f(x)= -\infty$ (from right side)

$\displaystyle \lim_{ x \to 2^-} f(x)= +\infty$ (from left side)

$\displaystyle \lim_{x \to \pm \infty} f(x)= 0$

2 Answers 2

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For the activity around x=2 you need $(2-x)$ raised to an odd power in the denominator. To make $f(1)=0$ you need $(x-1)$ in the numerator. And for the activity at plus and minus infinity you need a power of $x$ in the denominator. Thus I would suggest $f(x)=\frac{(x-1)}{(2-x)^3x}.$ Edit: Following comments below, a better suggestion is $f(x)=\frac{(x-1)}{(2-x)^3x^2}.$

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    Perfect! (I've already upvoted, else I'd upvote again).2011-11-03
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$(-\infty, 0)$ use $-1/x $

$(0,1]$ use $\ln(x)$

$[1,2)$ use $\ln(2-x)$

$(2,\infty)$ use $1/(x-2)$

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    There are a few issues with this answer. For starters, I think the question is asking for one rational function which has all these properties. Second, $\ln(x)$ is not a rational function.2011-11-03