This gets a little involved. Define *f* by

For we can find *f*'(*x*) by using the standard rules: it's
. This doesn't have
a limit as *x* goes to 0: if you look at the sequence
(which goes to 0), , but looking at the
sequence , .
Therefore does not exist.

However, *f*'(0) *does* exist. Use the definition of derivative:

Since , the difference quotient satisfies

for all . But , so

by the pinching theorem. Therefore *f*'(0) exists and equals 0.

Here's a picture of *f*:

Mon May 5 12:53:33 CDT 1997