determine whether the following statements are true and give an explanation or counterexample. d. false…

Explanation:

Step1: Recall rational - function limit rules

For a rational function $f(x)=\frac{P(x)}{Q(x)}=\frac{a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0}{b_mx^m + b_{m - 1}x^{m-1}+\cdots+b_0}$, where $a_n\neq0$ and $b_m\neq0$. If $n < m$, $\lim_{x\rightarrow\pm\infty}f(x)=0$. If $n = m$, $\lim_{x\rightarrow\pm\infty}f(x)=\frac{a_n}{b_m}$.

Step2: Analyze the given statement

If a rational function has a finite limit as $x\rightarrow\infty$, then either the degree of the numerator $n$ is less than the degree of the denominator $m$ ($n<m$) or the degree of the numerator is equal to the degree of the denominator ($n = m$). In both cases, $\lim_{x\rightarrow-\infty}f(x)$ is also finite. When $n<m$, $\lim_{x\rightarrow-\infty}f(x)=0$ and when $n = m$, $\lim_{x\rightarrow-\infty}f(x)=\frac{a_n}{b_m}$, the same as $\lim_{x\rightarrow\infty}f(x)$.

Answer:

A. True. If a rational function has a finite limit as $x\rightarrow\infty$, then either the degree of the numerator is the same as the degree of the denominator or the numerator is less than the degree of the denominator. In either of these instances, the limit as $x\rightarrow-\infty$ is finite.