How do you find the vertical, horizontal or slant asymptotes for #f(x) = x/((x+3)(x-4))#?

1 Answer
May 7, 2018

Answer:

Vertical Asymptotes are based on any factors in the Denominator while Horizontal/Slant Asymptote is based on the highest power of #x# in the Numerator and Denominator.

Explanation:

Vertical Asymptote:
Solve for #x# for each factor in the Denominator.
#x + 3 = 0# and #x - 4 = 0#
So Vertical Asymptote will appear at #x = -3# and #x = 4#

Horizontal Asymptote:
Let's say #f(x) = (ax^n)/(bx^m)#

If #n = m#, then Horizontal Asymptote is #y = a/b# (simplified).

If #n < m#, then Horizontal Asymptote is #y = 0#.

If #n > m#, then Horizontal Asymptote is None because it doesn't exist. Slant Asymptote only occurs if #n = m + 1#. We would have to use Long Division or Synthetic Division to find the Linear Slant Asymptote.

For the problem above, multiply the bottom factors:
#x/(x^2 -x - 12)#
Top Power of 1 #<# Bottom Power of 2.
So Horizontal Asymptote is #y = 0#