How do you determine the vertical and horizontal asymptotes of the graph of the function?

To find the vertical asymptote, you set the denominator equal to zero, although, there are 3 parts to this equation 6, #14/(4x+36)#, and #1/(5x^4)#

To set the denominator to zero, they all need to have equal denominators

I would start by making #14/(4x+36)# and 6 have the same denominator. So, 6 (which currently has a denominator of 1) needs a denominator of #4x+36#

#6/1 xx (4x+36)/(4x+36)= (24x + 216)/(4x+36)#

Now the equation should look like:

#y=-(24x + 216)/(4x+36) + 1/(5x^4)#

(Remember #-(24x + 216)/(4x+36)# is negative because in the original equation, #14/(4x+36)# was negative)

Next, #-(24x + 216)/(4x+36)# and #1/(5x^4)# need the same denominator

#(color(red)((5x^4)/(5x^4))# #xx# #-(24x + 216)/(4x+36))# + #(1/(5x^4)# #xx# #color(green)((4x+36)/(4x+36)))#

#color(red)(-(120x^5+1080x^4)/(20x^5+180x^4))# + #color(green)((4x+36)/(20x^5+180)#

Now, as you can see, they both have the same denominator, and by doing so, they are now easy to combine the numerators because you are adding fractions with like denominators

(The negative on the first fraction (written in red) can be distributed)

#y=(120x^5-1080x^4+4x+36)/(20x^5+180x^4)#

Now, since the equation is one large fraction, you can now set the denominator equal to zero to get what the vertical asymptotes are and factor

#20x^5+180x^4=0#

#20x^4(20x+180)=0#

#20x^4=0#

#x=0#

#20x+180=0#

#20x=-180#

#x=-9#

This means there is a vertical asymptote at #x=0# and #x=-9#

To find the horizontal asymptote, you have to look at the biggest power of the numerator and denominator

(#:.# means therefore)
(HA is horizontal asymptote)
(#EE# means there exists)

• If numerator > denominator, there is no HA

#y=(x^3)/(x^2)# #:.# #EE# no HA

• If numerator < denominator, HA is y=0

#y=x/x^2# #:.# #EE# HA at y=0

• If numerator = denominator, look at the leading coefficients

#y=(color(red)5x^3+7)/(color(red)2x^3-8)# #:.# #EE# HA at #y=5/2#

#y=(color(red)120color(red)x^color(red)5-1080x^4+4x+36)/(color(red)20color(red)x^color(red)5+180x^4)#

The exponents of the numerator and the denominator are equal, so take the leading coefficients of each

#y=120/20#

Simplified:

#y=6#

HA at #y=6#