find all horizontal asymptotes of the following function.\n\n$f(x)=\\frac{2(x + 2)(x - 2)}{3(2x - 3)(x + 3)}$
find all horizontal asymptotes of the following function.\n\n$f(x)=\\frac{2(x + 2)(x - 2)}{3(2x - 3)(x + 3)}$
Answer
Explanation:
Step1: Expand numerator and denominator
Expand (2(x + 2)(x - 2)=2(x^{2}-4)=2x^{2}-8) Expand (3(2x - 3)(x + 3)=3(2x^{2}+6x-3x - 9)=3(2x^{2}+3x - 9)=6x^{2}+9x - 27) So (f(x)=\frac{2x^{2}-8}{6x^{2}+9x - 27})
Step2: Use the rule for horizontal asymptotes of rational functions
For a rational function (y = \frac{f(x)}{g(x)}=\frac{a_{n}x^{n}+\cdots+a_{0}}{b_{m}x^{m}+\cdots+b_{0}}), if (n = m), the horizontal asymptote is (y=\frac{a_{n}}{b_{m}}) Here (n = 2), (m = 2), (a_{2}=2), (b_{2}=6)
Answer:
The horizontal asymptote is (y=\frac{1}{3})