How do you find all the asymptotes for function $f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)$ ?

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Answer 1 · 2016-02-26T10:54:18

$x=-root(5)6$ and $x=-root(5)5$ are two vertical asymptotes. Horizontal asymptote is $y=11$ .

To find asymptotes of $f(x)=(11(x−5)^5*(x+6)^5)/(x^10+11x^5+30)$ , we should factorize denominator $(x^10+11x^5+30)$

The factors of $(x^10+11x^5+30)$ are $(x^5+5)$ and $(x^5+6)$

As $(x+a)$ is a factor of $(x^5+a^5)$ (as $-a$ is zero of latter)

two factors of $(x^10+11x^5+30)$ are $x=-root(5)6$ and $x=-root(5)5$

and hence $x=-root(5)6$ and $x=-root(5)5$ are two vertical asymptotes.

It is also apparent that the highest degree of numerator is $11x^10$ and that of denominator is $x^10$ . As their ratio is $11$

Horizontal asymptote is $y=11$ .

How do you find all the asymptotes for function f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)?