Question

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

Explanation:

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

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

Step2: Apply limit properties

As \(x
ightarrow\pm\infty\), \(\lim_{x
ightarrow\pm\infty}\frac{1}{x^{n}} = 0\) for \(n>0\). So

$$\begin{align*} \lim_{x ightarrow\pm\infty}\frac{7+\frac{3}{x^{2}}+\frac{1}{x^{4}}}{3-\frac{5}{x^{3}}-\frac{2}{x^{4}}}&=\frac{\lim_{x ightarrow\pm\infty}(7 + 0+0)}{\lim_{x ightarrow\pm\infty}(3-0 - 0)}\\ &=\frac{7}{3} \end{align*}$$

Answer:

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