Question

Consider the following rational functions: r(x)=(2x−1)/(x^2)−x−2 s(x)=(x^3+27)/(x^2+4) t(x)=((x^3)−9x)/(x+2) u(x)=((x^2)+x−6)/(x^2−25) w(x)=((x^3)+(6x^2)+9x))/(x+3) Which of these functions has no vertical asymptote?

Rational functions
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asked 2021-05-26
Consider the following rational functions: \(\displaystyle{r}{\left({x}\right)}=\frac{{{2}{x}−{1}}}{{{x}^{{2}}}}−{x}−{2}{s}{\left({x}\right)}=\frac{{{x}^{{3}}+{27}}}{{{x}^{{2}}+{4}}}\)
\(\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}\)
\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{x}^{{2}}−{25}}}\)
\(\displaystyle{w}{\left({x}\right)}={\left({\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}\right)}\frac{{)}}{{{x}+{3}}}\)
Which of these functions has no vertical asymptote?

Answers (1)

2021-05-27
\(r(x)=(2x−1)/((x^2)−x−2 )= (2x-1)/(x-2)(x+1)\)
\(s(x)=(x^3+27)/(x^2+4)=(x+3)((x^2)+9+3x)/((x^2)+4)\)
\(t(x)=((x^3)−9x)/(x+2)=(x(x-3)(x+3))/(x+2)\)
\(u(x)=((x^2)+x−6)/(x^2−25)=((x-2)(x+3))/((x-5)(x+5))\)
\(w(x)=((x^3)+(6x^2)+9x))/(x+3)=x(x+3)\)
s(x) and w(x)
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