The idea of an asymptote is best illustrated by examples.
Example: Consider the following rational function
Its domain is all x's different from 2. At x=2 the denominator is equal to 0, and the function is not defined. What happens for x's near 2? Near a point where the donominator is 0 strange things may happen. Let's examine the behavior of f(x) near 2 by looking at its values for x near 2, and then the graph y=f(x) near 2.