How do I find the asymptotes of #f(x)= (- 4x-x^2)/(2 +2x- x^3)#?

Question

Eli Assouline · Accepted Answer

asymptotes:
#x=2#
#x=sqrt(2)#
#x=-sqrt(2)# Start by simplifying the function: #f(x)=(-4x-x^2)/(2+2x-x^3)# #f(x)=(-x(4+x))/(-x^3+2x+2)# #f(x)=(-x(x+4))/(-x(x^2-2)+2)# #f(x)=(-x(x+4))/((-x+2)(x^2-2))# #f(x)=(-x(x+4))/(-(x-2)(x^2-2))# #f(x)=(color(red)cancelcolor(black)-x(x+4))/(color(red)cancelcolor(black)-(x-2)(x^2-2))# #f(x)=(x(x+4))/((x-2)(x^2-2))# Take each bracketed polynomial in the denominator, set it to cannot equal to #0#, and solve for #x#. Finding the asymptotes #1. x-2!=0# #color(white)(ixxxx)x!=2# #2. x^2-2!=0# #color(white)(xxxx)x^2!=2# #color(white)(xxxxx)x!=+-sqrt(2)# The asymptotes are also the values which cannot be substituted into the equation such that the denominator would be #0#. #:.#, the asymptotes are #x=2#, #x=sqrt(2)#, and #x=-sqrt(2)#.

Ethan Adkins · Answer

undefined To find the asymptotes of the function $ f(x) = \frac{-4x - x^2}{2 + 2x - x^3} $, first identify the vertical asymptotes by finding the values of $ x $ for which the denominator equals zero. Then, find the horizontal asymptotes by examining the behavior of the function as $ x $ approaches positive and negative infinity.

Vertical asymptotes occur when the denominator of the function is equal to zero. So, set the denominator $ 2 + 2x - x^3 $ equal to zero and solve for $ x $ to find the vertical asymptotes.

$$ 2 + 2x - x^3 = 0 $$

After solving the equation, you'll find the values of $ x $ that make the denominator zero. These values will be the vertical asymptotes.

Horizontal asymptotes are found by analyzing the behavior of the function as $ x $ approaches positive and negative infinity. To find horizontal asymptotes, consider the degrees of the numerator and denominator polynomials.

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $.

If the degree of the numerator equals the degree of the denominator, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

After identifying both the vertical and horizontal asymptotes, you will have a comprehensive understanding of the behavior of the function $ f(x) $.