Question

suppose that f(x) and g(x) are polynomials in x and that lim(x→∞) f(x)/g(x)=2. can you conclude anything about lim(x→ - ∞) f(x)/g(x)? give reasons for your answer. choose the correct choice below. a. it is possible to conclude that lim(x→ - ∞) f(x)/g(x)=2, because if lim(x→∞) f(x)/g(x)=2, then the polynomials f(x) and g(x) must have the same degree. thus, it is possible to divide the numerator and denominator by the largest x - term and have a constant remaining. b. there is not enough information to conclude anything about the lim(x→ - ∞) f(x)/g(x). c. the lim(x→ - ∞) f(x)/g(x) does not exist, because g(x) is zero as x approaches negative infinity. d. it is possible to conclude that lim(x→ - ∞) f(x)/g(x)= - 2, because if lim(x→∞) f(x)/g(x)=2, then lim(x→ - ∞) f(x)/g(x) is the opposite of lim(x→∞) f(x)/g(x).

Explanation:

Step1: Recall polynomial limit rules

For polynomials \(f(x)=a_nx^n+\cdots+a_0\) and \(g(x)=b_mx^m+\cdots + b_0\), \(\lim_{x
ightarrow\pm\infty}\frac{f(x)}{g(x)}\) depends on the degrees \(n\) and \(m\) of \(f(x)\) and \(g(x)\). If \(\lim_{x
ightarrow\infty}\frac{f(x)}{g(x)} = L
eq0,\pm\infty\), then \(n = m\).

Step2: Analyze the given limit

Given \(\lim_{x
ightarrow\infty}\frac{f(x)}{g(x)}=2\), the degrees of \(f(x)\) and \(g(x)\) are the same. When \(x
ightarrow-\infty\), for polynomials of the same - degree \(n\), \(\lim_{x
ightarrow-\infty}\frac{f(x)}{g(x)}=\lim_{x
ightarrow-\infty}\frac{a_nx^n+\cdots+a_0}{b_nx^n+\cdots + b_0}\). Dividing both the numerator and denominator by \(x^n\), we get \(\lim_{x
ightarrow-\infty}\frac{a_n+\frac{a_{n - 1}}{x}+\cdots+\frac{a_0}{x^n}}{b_n+\frac{b_{n - 1}}{x}+\cdots+\frac{b_0}{x^n}}\). As \(x
ightarrow-\infty\), \(\frac{a_i}{x^k}
ightarrow0\) for \(k\gt0\) and \(i\geq0\). So \(\lim_{x
ightarrow-\infty}\frac{f(x)}{g(x)}=\lim_{x
ightarrow\infty}\frac{f(x)}{g(x)} = 2\).

Answer:

A. It is possible to conclude that \(\lim_{x
ightarrow-\infty}\frac{f(x)}{g(x)} = 2\), because if \(\lim_{x
ightarrow\infty}\frac{f(x)}{g(x)}=2\), then the polynomials \(f(x)\) and \(g(x)\) must have the same degree. Thus, it is possible to divide the numerator and denominator by the largest \(x\) - term and have a constant remaining.