Given the function f(x) = ( Absolute Value of (3x -1) )/(x -1), shown above is the graph of f(x) using the Mathematica command Plot[ ]. Notice that f(x) is undefined when x = 1 and the line x = 1 is a vertical asymptote. (Note: Click on the image to see a larger view.)
As x approaches 1 from the left, f(x) approaches negative infinity.
As x approaches 1 from the right, f(x) approaches positive infinity.
For a limit to exist, the limit from the left must be equal to the limit from the right.
Therefore, we can say that the limit of f(x) as x approaches 1 does not exist (DNE) .
Note: I gave a similar item to my Ma 20 (Calculus) class in their first long exam today (1 July 2008). The students do not have an access to a computer or a graphing calculator during the exam. But they have access to a scientific calculator. So the students have to plot f(x) by producing an x y table where y = f(x). For example, when x = 0, y = -1. This item is worth 10 points in the exam. There were 10 items with a total of 100 points.
(P.S. Thanks for the feedback/correction I received from Dr. Pablo Manastas.)