One method has already been seen: an approaching table of values (see "Definition of a Limit"). This method can be used in any scenario, but will not be as precise as some other methods.
DIRECT SUBSTITUTION: This is for those scenarios where the value equals the limit (see "Limits and Explicit Values"). One simply evaluates "the limit of f(x) as x approaches c" as "the value of f(x) at x=c".
ELIMINATION OF HOLES: A function with a hole looks, to the naked eye, like a continuous function; however, there is one spot where it is empty - a hole. This hole usually occurs when the denominator would be zero; rendering the use of direct substitution impossible. Instead of a table of values, one can eliminate the hole through factorization and the cancellation of common factors (Algebra II), and then use substitution of the approaching value.
GRAPHING: One can use a graphing calculator or grapher application to graph the function, and trace along the function, approaching the value of limit and "eyeballing" the result.