question\nfind the horizontal asymptote for the function given below. give your answer in the form ( y = b…

question\nfind the horizontal asymptote for the function given below. give your answer in the form ( y = b ), or enter ( varnothing ) if there is none.\n f(x)=\frac{7 x^{4}+3 x^{2}+1}{3 x^{4}-5 x - 2} \nprovide your answer below:

question\nfind the horizontal asymptote for the function given below. give your answer in the form ( y = b ), or enter ( varnothing ) if there is none.\n f(x)=\frac{7 x^{4}+3 x^{2}+1}{3 x^{4}-5 x - 2} \nprovide your answer below:

Answer

Explanation:

Step1: Divide numerator and denominator by (x^4)

$$ \begin{align*} \lim_{x\rightarrow\pm\infty}f(x)&=\lim_{x\rightarrow\pm\infty}\frac{7x^{4}+3x^{2}+1}{3x^{4}-5x - 2}\ &=\lim_{x\rightarrow\pm\infty}\frac{7+\frac{3}{x^{2}}+\frac{1}{x^{4}}}{3-\frac{5}{x^{3}}-\frac{2}{x^{4}}} \end{align*} $$

Step2: Apply limit properties

As (x\rightarrow\pm\infty), (\lim_{x\rightarrow\pm\infty}\frac{1}{x^{n}} = 0) for (n>0). So $$ \begin{align*} \lim_{x\rightarrow\pm\infty}\frac{7+\frac{3}{x^{2}}+\frac{1}{x^{4}}}{3-\frac{5}{x^{3}}-\frac{2}{x^{4}}}&=\frac{\lim_{x\rightarrow\pm\infty}(7 + 0+0)}{\lim_{x\rightarrow\pm\infty}(3-0 - 0)}\ &=\frac{7}{3} \end{align*} $$

Answer:

(y=\frac{7}{3})