You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
is not defined if |x| > 1 but it is defined if |x| <= 1.
Let f(x) denote the price of x units of the product described above.
Notice that something very peculiar happens when the quantity crosses the value 1,000. The price for 999 items is $1,998 but the price for 1,000 items is only $1,500. By buying one additional item the cost drops $448.00.
This kind of behavior, although mathematically strange, is not uncommon. The function f(x) behaves differently on the two sides of the value x = 1,000.
In this kind of situation we use the notation
to describe what happens to the function f(x) when x is close to a but to the right of a, and the notation
to describe what happens to the function f(x) when x is close to a but to the left of a.
In our example above
These limits are called one-sided limits because each one looks at just one side of the value x = a.
For each of the following functions find the two one-sided limits at the indicated point a.
