How do you find the vertical, horizontal or slant asymptotes for f(x) = x/((x+3)(x-4))f(x)=x(x+3)(x4)?

1 Answer
May 7, 2018

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

Explanation:

Vertical Asymptote:
Solve for xx for each factor in the Denominator.
x + 3 = 0x+3=0 and x - 4 = 0x4=0
So Vertical Asymptote will appear at x = -3x=3 and x = 4x=4

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

If n = mn=m, then Horizontal Asymptote is y = a/by=ab (simplified).

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

If n > mn>m, then Horizontal Asymptote is None because it doesn't exist. Slant Asymptote only occurs if n = m + 1n=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)xx2x12
Top Power of 1 << Bottom Power of 2.
So Horizontal Asymptote is y = 0y=0