the rational function f is given by f(x) = ((3x - 5)(2x + 9)(x - 1))/((2x+1)(x - 6)(x + 7)) and is…

the rational function f is given by f(x) = ((3x - 5)(2x + 9)(x - 1))/((2x+1)(x - 6)(x + 7)) and is equivalent to f(x) = p(x)/q(x), where p, q are polynomial functions. which of the following statements about the end - behavior is true? lim_(x→∞) f(x)=? a the degree of p is greater than the degree of q, so the limit is 0. b the degree of p is greater than the degree of q, so the limit is ∞. c the degree of p is equal to the degree of q, so the limit is 3. d the degree of p is equal to the degree of q, so the limit is 1/3.

the rational function f is given by f(x) = ((3x - 5)(2x + 9)(x - 1))/((2x+1)(x - 6)(x + 7)) and is equivalent to f(x) = p(x)/q(x), where p, q are polynomial functions. which of the following statements about the end - behavior is true? lim_(x→∞) f(x)=? a the degree of p is greater than the degree of q, so the limit is 0. b the degree of p is greater than the degree of q, so the limit is ∞. c the degree of p is equal to the degree of q, so the limit is 3. d the degree of p is equal to the degree of q, so the limit is 1/3.

Answer

Explanation:

Step1: Determine degrees of numerator and denominator polynomials

The numerator (p(x)=(3x - 5)(2x+9)(x - 1)=6x^{3}+11x^{2}-52x + 45), degree of (p(x)) is (3). The denominator (q(x)=(2x + 1)(x-6)(x + 7)=2x^{3}-3x^{2}-89x-42), degree of (q(x)) is (3).

Step2: Use limit rule for rational functions

For a rational function (f(x)=\frac{p(x)}{q(x)}) where (\text{deg}(p)=\text{deg}(q)=n), (\lim_{x\rightarrow\infty}f(x)=\frac{a_{n}}{b_{n}}), where (a_{n}) and (b_{n}) are the leading - coefficients of (p(x)) and (q(x)) respectively. Here (a_{n}=6) and (b_{n}=2), so (\lim_{x\rightarrow\infty}f(x)=\frac{6}{2}=3).

Answer:

C. The degree of (p) is equal to the degree of (q), so the limit is (3)