suppose that f(x) and g(x) are polynomials in x and that lim(x→∞) f(x)/g(x)=2. can you conclude anything…

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).

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).

Answer

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\rightarrow\pm\infty}\frac{f(x)}{g(x)}) depends on the degrees (n) and (m) of (f(x)) and (g(x)). If (\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)} = L\neq0,\pm\infty), then (n = m).

Step2: Analyze the given limit

Given (\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=2), the degrees of (f(x)) and (g(x)) are the same. When (x\rightarrow-\infty), for polynomials of the same - degree (n), (\lim_{x\rightarrow-\infty}\frac{f(x)}{g(x)}=\lim_{x\rightarrow-\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\rightarrow-\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\rightarrow-\infty), (\frac{a_i}{x^k}\rightarrow0) for (k\gt0) and (i\geq0). So (\lim_{x\rightarrow-\infty}\frac{f(x)}{g(x)}=\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)} = 2).

Answer:

A. It is possible to conclude that (\lim_{x\rightarrow-\infty}\frac{f(x)}{g(x)} = 2), because if (\lim_{x\rightarrow\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.