Limits of Complex Functions at x = 0

[majesticman]

Hey ppl...

Is x/x differentiable at x = 0

Now i know that it is not defined at x=0 but the function does approach the same limits from either side...From what i remember the limit does exist (what was the name of the rule that let's you do that)...but does that mean it is differentiable at x= 0 ?

Also can the same idea be extended to complex functions (say z/z)

Thanks in advance

 

[HallsofIvy]

x/x??

As you say, that is not defined at x= 0 and so cannot be continuous or differentiable there.

Of course, it has a "removable" discontinuity at x= 0. For all x other than 0, x/x= 1 so the limit, as x goes to 0, is 1. We can make this function continuous at x= 0 by defining it to be 1 there. In that case, we just have the function f(x)= 1 for all x. It's derivative is the constant 0.

But what does this have to do with complex numbers? Did you mean to ask about z/|z| ? That would be a much more interesting question!