Understanding Complex Numbers: Limits & Functions

[chaoseverlasting]

[SOLVED] Complex Numbers

A very happy new year to all at PF.

Homework Statement

Kreyszig, P.665 section 12.3,

A function f(z) is said to have the limit l as z approaches a point $$z_0$$ if f is defined in the neighborhood of $$z_0$$ (except perhaps at $$z_0$$ itself) and if the values of f are "close" to l for all z "close" to $$z_0$$; that is, in precise terms, for every positive real $$\epsilon$$ we can find a positive real $$\delta$$ such that for all z not equal to $$z_0$$ in the disk $$|z-z_0|<\delta$$, we have

$$|f(z)-l|<\epsilon$$...(2)

that is, for every z not equal to $$z_0$$ in that $$\delta$$ disk, the value of f lies in the disk (2).

I think this means that if you were to plot z and f(z), for all values of z near the point $$z_0$$ (within the $$\delta$$ disk), but not necessarily at $$z_0$$ itself (as the function may not exist at $$z_0$$), the value of f(z) would be very close to l (but not necessarily l, as $$f(z_0)$$ may not exist) and that this value of f(z) would be inside the $$\epsilon$$ disk.

Is that right?

 

[mjsd]

sounds good

 

[unplebeian]

Congrats, you just understood what a limit is.

 

[chaoseverlasting]

Thank you.