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