It should be no secret by now that for most
functions defined by a single formula, when exists. For more complicated functions, this
may not be true. For dealing with some of these exceptional cases, we
need to define right-hand and left-hand limits. Loosely speaking, the
right-hand limit of at **a** is **L** if approaches **L** as
**x** approaches **a** from the right. That is, the values of **x** satisfy
**x > a**. The left-hand limit is defined in an analogous manner, with
the values of **x** approaching **a** from the left. Maple can
compute these special limits with commands like those shown below. The
Maple `floor` function is actually the greatest integer function.

> plot(floor(x),x=0..4);

> limit(floor(x),x=1,right);

> limit(floor(x),x=1,left);

The `floor` function is one of Maple's defined functions, so you
might expect things to work properly. If you define your own function
in a piecewise fashion, however, a different approach is needed.
For example, suppose you wanted to find the limit as **x** approaches
**2** for the following function:

You would need to execute the following commands:

> f1 := x -> sqrt(2-x);

> limit(f1(x),x=2,left);

> f2 := x -> x;

> limit(f2(x),x=2,right);

From this, we see that the limit of **f** as **x** approaches **2** does not exist since the limit from the left does not equal the limit from the right.

Thu Sep 7 15:27:05 EDT 1995