Limits of Complex Functions at x = 0

In summary, the conversation discusses the differentiability of the function x/x at x = 0 and whether the same idea can be applied to complex functions. The function is not defined at x = 0, but has a "removable" discontinuity. The limit of the function as x approaches 0 is 1, and the function can be made continuous by defining it to be 1 at x = 0. The conversation also mentions the derivative of the function and the potential application to complex numbers.
  • #1
majesticman
26
0
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
 
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  • #2
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!
 

1. What are the types of limits for complex functions?

The three types of limits for complex functions are finite limits, infinite limits, and limits at infinity. Finite limits occur when the limit of a function approaches a single finite value. Infinite limits occur when the limit of a function approaches positive or negative infinity. Limits at infinity occur when the limit of a function approaches a finite value as the input of the function approaches infinity.

2. How do you calculate the limit of a complex function?

The limit of a complex function can be calculated by plugging in the value that the function is approaching into the function and solving for the resulting expression. If the resulting expression is undefined, then other methods such as L'Hopital's rule or factoring can be used to evaluate the limit.

3. Can a complex function have multiple limits?

Yes, a complex function can have multiple limits. This can occur when the function has different limits as the input approaches from different directions. For example, the limit of a function may approach a different value from the left side compared to the right side.

4. What is a removable discontinuity in the context of complex functions?

A removable discontinuity occurs when a function has a hole or gap in its graph at a specific point. This means that the limit of the function exists at that point, but the function is not defined at that point. The discontinuity can be removed by defining the function at that point, resulting in a continuous function.

5. How do limits of complex functions relate to analytic functions?

Analytic functions are complex functions that can be represented by a Taylor series, and their derivatives exist at all points within their domain. Limits of complex functions are used to determine the behavior of analytic functions at specific points and to confirm their continuity. Analytic functions must have finite limits at all points within their domain.

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