How do you find vertical, horizontal and oblique asymptotes for #G(x)=(3*x^4 +4)/(x^3+3*x)#?

Question

Christian Antunez · Accepted Answer

Vertical asymptote at #x_v = 0#
Slant asymptote #y = 3x#

In a polynomial fraction #f(x) = (p_n(x))/(p_m(x))# we have:

#1)# vertical asymptotes for #x_v# such that #p_m(x_v)=0#
#2)# horizontal asymptotes when #n le m#
#3)# slant asymptotes when #n = m + 1#
In the present case we have #x_v =0# and #n = m+1# with #n = 4# and #m = 3#

Slant asymptotes are obtained considering

#(p_n(x))/(p_{n-1}(x)) approx y = a x+x#

for large values of #abs(x)#.
In the present case we have

#(p_n(x))/(p_{n-1}(x)) = (3x^4+4)/(x^3+3x)#

then

#p_n(x)=p_{n-1}(x)(a x+b)+r_{n-2}(x)#
#r_{n-2}(x)=c x^2 + d x+e#
# (3x^4+4) = (x^2-4)(a x + b) + c x^2 + d x + e#

equating coefficients

#{ (4 - e=0), (-3 b - d=0),( -3 a - c=0),( -b=0),(3 - a=0) :}#

solving for #a,b,c,d,e# we have #{a =3, b = 0, c = -9, d = 0,e=4}#
substituting in #y = a x + b#

#y = 3x #

Theodore Amidon · Answer

undefined To find the vertical asymptotes of a rational function, determine the values of $ x $ that make the denominator zero, excluding any values that would make the numerator zero as well. For horizontal asymptotes, compare the degrees of the numerator and the denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is $ y = 0 $. For oblique asymptotes, divide the numerator by the denominator using long division or synthetic division. The oblique asymptote is the quotient obtained from the division.

For $ G(x) = \frac{3x^4 + 4}{x^3 + 3x} $:

1. Vertical asymptotes: Set the denominator equal to zero and solve for $ x $. $ x^3 + 3x = 0 $ has a solution of $ x = 0 $.

2. Horizontal asymptote: The degrees of the numerator and denominator are equal, so we compare the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is $ y = \frac{3}{1} = 3 $.

3. Oblique asymptote: Perform polynomial division (long division or synthetic division) of $ 3x^4 + 4 $ by $ x^3 + 3x $. The quotient obtained from this division represents the oblique asymptote, if it exists.