Question

give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational function. f(x) = (4x^2 + 1)/(x + 2) select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the equation of the vertical asymptote is x = - 2. (type an equation.) b. there is no vertical asymptote. select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the equation of the horizontal asymptote is. (type an equation.) b. there is no horizontal asymptote. select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the equation of the oblique asymptote is. (type an equation.) b. there is no oblique asymptote.

Explanation:

Step1: Find vertical asymptote

Set denominator equal to 0. For $f(x)=\frac{4x^{2}+1}{x + 2}$, we solve $x+2=0$, so $x=-2$.

Step2: Determine horizontal asymptote

Degree of numerator ($n = 2$) is greater than degree of denominator ($m=1$), so there is no horizontal asymptote.

Step3: Find oblique asymptote

Perform polynomial long - division of $4x^{2}+1$ by $x + 2$. $4x^{2}+1=(x + 2)(4x-8)+17$. So oblique asymptote is $y = 4x-8$.

Answer:

A. The equation of the vertical asymptote is $x=-2$.
B. There is no horizontal asymptote.
A. The equation of the oblique asymptote is $y = 4x-8$.