When thinking of an asymptote, we are really thinking of a limit. For example, suppose we want the asymptote of the function
f(x) = 1/x, a very basic function but used to illustrate this point.
First we know that x cannot equal 0 because a function is undefined with a 0 denominator. So as x approaches 0, the function is undefined.
As x gets larger approching infinity, notice 1/x gets smaller
x = 10, 1/x = .1
x = 100, 1/x = .01
x = 10000, 1/x = .0001
Therefore 1/x approaches 0 as x approaches infinity
When x is very small and negative, 1/x approaches infinity as it does when x is very small and positive.
x = .1, 1/x = 10
x = .01, 1/x = 100
x = .0001, 1/x = 10000
x = -.1, 1/x = -10
x = -.01, 1/x = -100
x = -.0001, 1/x = -10000
As x values get more ad more negative, 1/x approaches 0.
x= -10, 1/x = -.1
x= -100, 1/x = -.01
x= -10000, 1/x = -.0001
The asymptotes are the lines x = 0 and y = 0.
The idea of a limit is a topic seen in precalculus and in calculus.
Lim 1/x = infinity
x-> 0 (positive)
This means the limit as x approaches 0 from the positive side equals infinity
Lim 1/x = -infinity
x -> 0 (negative)
The limit as x approaches 0 from the negative side equals negative infinity.
Lim 1/x = 0 and Lim 1/x = 0
x -> infinity x-> negative infinity