find the horizontal asymptote of ( f(x)=\frac{-3 x^{5}-4 x^{3}+5 x^{2}+2 x}{2 x^{5}+x^{4}-3 x} ). if the…

Explanation:

Step1: Divide numerator and denominator by (x^5)

$$ \begin{align*} \lim_{x\rightarrow\pm\infty}\frac{-3x^{5}-4x^{3}+5x^{2}+2x}{2x^{5}+x^{4}-3x}&=\lim_{x\rightarrow\pm\infty}\frac{-3-\frac{4}{x^{2}}+\frac{5}{x^{3}}+\frac{2}{x^{4}}}{2+\frac{1}{x}-\frac{3}{x^{4}}}\ \end{align*} $$

Step2: Evaluate the limit

As (x\rightarrow\pm\infty), (\frac{1}{x}\rightarrow0), (\frac{1}{x^{2}}\rightarrow0), (\frac{1}{x^{3}}\rightarrow0), (\frac{1}{x^{4}}\rightarrow0)

So (\lim_{x\rightarrow\pm\infty}\frac{-3-\frac{4}{x^{2}}+\frac{5}{x^{3}}+\frac{2}{x^{4}}}{2+\frac{1}{x}-\frac{3}{x^{4}}}=\frac{-3 - 0+0 + 0}{2+0 - 0})

Answer:

(-\frac{3}{2})