How do you find asymptotes?

Veritcal asymptotes are where the denominator of a fraction becomes 0 and the value of f(x) becomes undefined. Set the denominator to 0 and solve.


Horizontal asymptotes are values of f(x) when x→∞ and x→-∞

Two examples:
f(x)=3x² / (x²+1)

Vertical:
Set (x²+1)=0 and solve for x.
x²=-1 has no answers, so there is no vertical asymptote.

Horizontal:
Divide all terms by the highest power of x to eliminate unimportant values
dividing by x², you get 3 / (1 + (1/x²))
As x→∞ then 1/x² vanishes, leaving 3/1=3, so there is an asymptote at y=3

f(x)=(x-3) / (x²+3x)

Vertical:
Set (x²+3x)=0 and solve. x={0,-3} so there are vertical asymptotes at 0 and -3

Horizontal:
Divide out by x²
(x/x² - 3/x²) / (x²/x² + 3x/x²) as x→∞
The terms in the numerator all vanish, making the answer 0, so there is a horizontal asymptote at 0.

Enjoy.■
A vertical asymptote also exists for the value of x (assuming a function of x) where the function becomes undefined. For instance:
f(x) = ln (x). The logarithmic function is not defined for x<=0.
Ex: f(x) = ln (x+2)... There would be a vertical asymptote at x= -2.