determine the horizontal asymptote of the function. if none exists, state that fact.\n\n f(x)=\frac{3…

determine the horizontal asymptote of the function. if none exists, state that fact.\n\n f(x)=\frac{3 x^{3}-3 x + 4}{15 x^{3}+3 x - 7} \n\nselect the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\n\na. the function has one horizontal asymptote, (type an equation )\nb. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is (type equations )\nc. the function has no horizontal asymptotes.

determine the horizontal asymptote of the function. if none exists, state that fact.\n\n f(x)=\frac{3 x^{3}-3 x + 4}{15 x^{3}+3 x - 7} \n\nselect the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\n\na. the function has one horizontal asymptote, (type an equation )\nb. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is (type equations )\nc. the function has no horizontal asymptotes.

Answer

Explanation:

Step1: Divide numerator and denominator by (x^3)

$$\lim_{x\rightarrow\pm\infty}\frac{3x^{3}-3x + 4}{15x^{3}+3x - 7}=\lim_{x\rightarrow\pm\infty}\frac{3-\frac{3}{x^{2}}+\frac{4}{x^{3}}}{15+\frac{3}{x^{2}}-\frac{7}{x^{3}}}$$

Step2: Apply limit rules

As (x\rightarrow\pm\infty), (\frac{1}{x^{n}}\rightarrow0) for (n>0). So, (\lim_{x\rightarrow\pm\infty}\frac{3-\frac{3}{x^{2}}+\frac{4}{x^{3}}}{15+\frac{3}{x^{2}}-\frac{7}{x^{3}}}=\frac{3 - 0+0}{15+0 - 0})

Answer:

A. The function has one horizontal asymptote, (y = \frac{1}{5})