Question

determine the horizontal asymptote(s) of the following function. enter each asymptote as a separate equation, separating multiple asymptotes with a comma. if there are no horizontal asymptotes, enter none.

$$ f ( x ) = \frac { 5 x ^ { 2 } - 4 x + 6 } { 2 x ^ { 2 } - x + 3 } $$

Explanation:

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

$$\begin{align*} \lim_{x ightarrow\pm\infty}f(x)&=\lim_{x ightarrow\pm\infty}\frac{5x^{2}-4x + 6}{2x^{2}-x + 3}\\ &=\lim_{x ightarrow\pm\infty}\frac{5-\frac{4}{x}+\frac{6}{x^{2}}}{2-\frac{1}{x}+\frac{3}{x^{2}}} \end{align*}$$

Step2: Evaluate the limit

As \(x
ightarrow\pm\infty\), \(\frac{1}{x}
ightarrow0\) and \(\frac{1}{x^{2}}
ightarrow0\). So \(\lim_{x
ightarrow\pm\infty}\frac{5-\frac{4}{x}+\frac{6}{x^{2}}}{2-\frac{1}{x}+\frac{3}{x^{2}}}=\frac{5 - 0+0}{2-0 + 0}=\frac{5}{2}\)

Answer:

\(y = \frac{5}{2}\)