find all horizontal asymptotes of the following function.\n\n f(x)=\frac{5 x^{2}-27 x + 10}{2 x^{2}-8 x…

Explanation:

Step1: Divide numerator and denominator by (x^{2})

$$ \begin{align*} \lim_{x\rightarrow\pm\infty}f(x)&=\lim_{x\rightarrow\pm\infty}\frac{5x^{2}-27x + 10}{2x^{2}-8x - 10}\ &=\lim_{x\rightarrow\pm\infty}\frac{5-\frac{27}{x}+\frac{10}{x^{2}}}{2-\frac{8}{x}-\frac{10}{x^{2}}} \end{align*} $$

Step2: Evaluate the limit

As (x\rightarrow\pm\infty), (\frac{1}{x}\rightarrow0) and (\frac{1}{x^{2}}\rightarrow0) $$ \lim_{x\rightarrow\pm\infty}\frac{5-\frac{27}{x}+\frac{10}{x^{2}}}{2-\frac{8}{x}-\frac{10}{x^{2}}}=\frac{5 - 0+0}{2-0 - 0}=\frac{5}{2} $$

Answer:

One Horizontal Asymptote (y = \frac{5}{2})