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$f(x) = -x^4 + 2x^3 - \frac{2x}{(x-2)^2},\ g(x) = 1 - x^2$

I'm trying to figure out where to begin with graphing the rational function for $f$. I also need to determine the vertical asymptote. How do I solve this?

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1 Answer 1

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For f you might expect a vertical asymptote at x=2, so you have to find the limits

$\lim\limits_{x\to 2^{-}}f(x)=\lim\limits_{x\to 2^{+}}f(x)=-\infty$. Now in order to graph the function, study the first derivative values of $f'(x)$ and see where the function is increasing or decreasing and determine the global maximums and turning points. After this you will get the figure below .

enter image description here

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