Write as many limit statements as you can about the function graphed at right as and .
Limits give information about the location of holes and asymptotes, because they are predicted values, not real values.
An Intuitive Definition of Limit
When you graph a function , most of the time you can guess what the value of, say, is by knowing the values of when is very close to . One way to think about this is to assume you have the graph for , except at . Can you make a reasonable accurate guess as to the value of ? If so, and this value is , we say that the limit of exists at and use the notation .
For example, if , and , it is reasonable to guess that and therefore .
You can also take one-sided limits using numbers less than (the notation is ) or greater than (the notation is ).
An important point is that does not need to equal .