find all horizontal asymptotes of the following function.\n\n f(x)=\frac{2(2 x+9)(x+4)}{3(x+4)} \n\nanswer…

find all horizontal asymptotes of the following function.\n\n f(x)=\frac{2(2 x+9)(x+4)}{3(x+4)} \n\nanswer attempt 1 out of 2\n\none horizontal asymptote

find all horizontal asymptotes of the following function.\n\n f(x)=\frac{2(2 x+9)(x+4)}{3(x+4)} \n\nanswer attempt 1 out of 2\n\none horizontal asymptote

Answer

Explanation:

Step1: Simplify the function

Cancel out the common factor ((x + 4)) in the numerator and denominator. [ \begin{align*} f(x)&=\frac{2(2x + 9)(x + 4)}{3(x + 4)}\ &=\frac{2(2x+9)}{3}\ &=\frac{4x+18}{3}\ &=\frac{4}{3}x + 6 \end{align*} ] Since after simplification (f(x)) is a linear function (y=\frac{4}{3}x + 6), and for a non - constant linear function (y = mx + b(m\neq0)), there is no horizontal asymptote. But wait, we made a mistake above.

Let's use the limit method. The formula for horizontal asymptote: if (y = f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}), then

  • If (n\lt m), (y = 0) is the horizontal asymptote.
  • If (n=m), (y=\frac{a_n}{b_m}) is the horizontal asymptote.
  • If (n\gt m), there is no horizontal asymptote.

First, expand the numerator: (2(2x + 9)(x + 4)=2(2x^2+8x+9x + 36)=4x^2+34x + 72) The denominator is (3(x + 4)=3x+12)

The degree of the numerator (n = 2) and the degree of the denominator (m = 1), (n>m)

Answer:

There is no horizontal asymptote.