Question

find all horizontal asymptotes of the following function.

f(x)=\frac{5 x^{2}-27 x + 10}{2 x^{2}-8 x - 10}

answer attempt 1 out of 2

Explanation:

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

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

Step2: Evaluate the limit

As \(x
ightarrow\pm\infty\), \(\frac{1}{x}
ightarrow0\) and \(\frac{1}{x^{2}}
ightarrow0\)

$$ \lim_{x ightarrow\pm\infty}\frac{5-\frac{27}{x}+\frac{10}{x^{2}}}{2-\frac{8}{x}-\frac{10}{x^{2}}}=\frac{5 - 0+0}{2-0 - 0}=\frac{5}{2} $$

Answer:

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