Question

find the horizontal asymptote, if any, of the graph of the rational function.
g(x) = \frac{12x^{2}}{6x^{2}+1}
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
○ a. the horizontal asymptote is □ (type an equation.)
○ b. there is no horizontal asymptote.

Explanation:

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

$$\lim_{x ightarrow\pm\infty}g(x)=\lim_{x ightarrow\pm\infty}\frac{12x^{2}}{6x^{2}+1}=\lim_{x ightarrow\pm\infty}\frac{\frac{12x^{2}}{x^{2}}}{\frac{6x^{2}}{x^{2}}+\frac{1}{x^{2}}}$$

Step2: Evaluate the limit

$$=\lim_{x ightarrow\pm\infty}\frac{12}{6 + 0}=\frac{12}{6}=2$$

Answer:

A. The horizontal asymptote is \(y = 2\)