What are the asymptotes of $f(x)=-x/((x-1)(3-x))$ ?

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Answer 1 · 2016-10-28T12:35:06

$f(x)$ has the following asymptotes:
Vertical at $x=1$ and $x=3$
Horizontal at $y=0$ as $x->oo$ and at $y=0$ as $x->-oo$

$f(x)=-x/((x-1)(3-x))$

We will have vertical asymptotes when the denominator is zero,
ie
$(x-1)(3-x) = 0 => x=1,3$

As $x->oo => f(x) ~-x/(x(-x))$ ,
ie $f(x) ~1/x->0^+$

Similarly, As $x->-oo => f(x) -> 0^-$ ,

$f(x)$ has the following asymptotes:
Vertical at $x=1$ and $x=3$
Horizontal at $y=0$ as $x->oo$ and at $y=0$ as $x->-oo$

graph{-x/((x-1)(3-x)) [-10, 10, -5, 5]}

What are the asymptotes of f(x)=-x/((x-1)(3-x)) f(x)=-x/((x-1)(3-x)) ?