consider the following rational function f. f...

# Explanation: ## Step1: Identify the degrees of numerator and denominator The numerator $9x^{4}-5x^{2}-x$ has degree $n = 4$ and the denominator $3x - 7$ has degree $m=1$. Since $n>m$, we consider the leading - terms of the numerator and denominator. The leading - term of the numerator is $9x^{4}$ and the leading - term of the denominator is $3x$. ## Step2: Simplify the ratio of leading - terms The ratio of the leading - terms is $\frac{9x^{4}}{3x}=3x^{3}$. ## Step3: Analyze the end - behavior as $x\to-\infty$ When $x\to-\infty$, for the function $y = 3x^{3}$, we know that if $x$ is a large negative number, say $x=-N$ where $N$ is a large positive number, then $y = 3(-N)^{3}=-3N^{3}\to-\infty$. So $f(x)\to-\infty$ as $x\to-\infty$. ## Step4: Analyze the end - behavior as $x\to\infty$ When $x\to\infty$, for the function $y = 3x^{3}$, if $x = N$ where $N$ is a large positive number, then $y = 3N^{3}\to\infty$. So $f(x)\to\infty$ as $x\to\infty$. # Answer: $f(x)\to-\infty$ as $x\to-\infty$; $f(x)\to\infty$ as $x\to\infty$