- #1
Genericcoder
- 131
- 0
So there is something I don't understand in the definition of limit that is applied to some problem
I have some intuition for like the rigorous limit definition but I don't have full understanding when applied to some problems.
Use definition 2 to prove lim as z → i of z^2 = -1
The book answer:
We must show that for given E > 0 there is a positive number x such that
|z^2 - (-1)| < E whenever 0 < |z - i| :
so we express |z^2 - (-1) | in terms of |z - i|:
z^2 - (-1) = z^2 + 1 = (z - i)(z + i) = (z - i)(z - i + 2i)
It follows from the properties of absolute value defived in Sec 1.3
that
|z^2 - (-1)| = |z - i||z - i + 2i| <= |z - i| (|z - i| + 2)
Now if |z - i| < x the right hand is less than x(x + 2) so to ensure that it is less than E, we can choose x to be smaller than either of the number E/3 and 1:
|z - i||(|z - i| + 2) < E/3(1 + 2) = 2
So Here there is a lot of stuff that I don't understand like why did we need to express one value in terms of the other ? I still don't know follow what's going on here if someone could explain please because I want full understanding of these stuff. thank you.
I have some intuition for like the rigorous limit definition but I don't have full understanding when applied to some problems.
Use definition 2 to prove lim as z → i of z^2 = -1
The book answer:
We must show that for given E > 0 there is a positive number x such that
|z^2 - (-1)| < E whenever 0 < |z - i| :
so we express |z^2 - (-1) | in terms of |z - i|:
z^2 - (-1) = z^2 + 1 = (z - i)(z + i) = (z - i)(z - i + 2i)
It follows from the properties of absolute value defived in Sec 1.3
that
|z^2 - (-1)| = |z - i||z - i + 2i| <= |z - i| (|z - i| + 2)
Now if |z - i| < x the right hand is less than x(x + 2) so to ensure that it is less than E, we can choose x to be smaller than either of the number E/3 and 1:
|z - i||(|z - i| + 2) < E/3(1 + 2) = 2
So Here there is a lot of stuff that I don't understand like why did we need to express one value in terms of the other ? I still don't know follow what's going on here if someone could explain please because I want full understanding of these stuff. thank you.