Question

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.

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 \(\text{deg}(p)=\text{deg}(q)=n\), \(\lim_{x
ightarrow\infty}f(x)=\frac{a_{n}}{b_{n}}\), 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 \(\lim_{x
ightarrow\infty}f(x)=\frac{6}{2}=3\).

Answer:

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