 | | From: | emailzul at starhub.net.sg | | Subject: | Solving complex limit using epsilon delta | | Date: | 23 Jan 2005 07:55:25 -0800 |
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 | Hi,
How do I prove
lim Z^n = Zo^n as Z tends to Zo?
I manage to do via induction by using a theorem which states that
lim [f(z)*F(z)] = woWo but is there a way to generalize the proof using
epsilon delta?
Cos I have managed to proof for Lim Z^2=Zo^2 as Z tends to Zo but
unable to do so for Lim Z^3=Zo^3 as Z tends to Zo.
I know I need to show there for every eps>0 there exist a delta> 0 such
that
|Z^3-Zo^3| < eps whenever 0<|Z-Zo|< delta
but my starting proof goes like this,
|f(Z)-Wo|=|Z^3-Zo^3| = (Z-Zo)*(Z^2 + Z*Zo + Zo^2)
then I got stuck.
I would appreciate a general proof so I could apply to higher powers of
Z.
Thank you in advance.
Regards
Zul
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| | From: | José_Carlos_Santos | | Subject: | Re: Solving complex limit using epsilon delta | | Date: | Sun, 23 Jan 2005 16:04:03 +0000 |
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| emailzul@starhub.net.sg wrote:
> I know I need to show there for every eps>0 there exist a delta> 0 such
> that
> |Z^3-Zo^3| < eps whenever 0<|Z-Zo|< delta
>
> but my starting proof goes like this,
>
> |f(Z)-Wo|=|Z^3-Zo^3| = (Z-Zo)*(Z^2 + Z*Zo + Zo^2)
>
> then I got stuck.
Suppose that |Z - Z_0| < 1. Then |Z| < 1 + |Z_0| and so
|Z^2 + Z*Z_0 + Z_0^2| <= (1 + |Z_0|)^2 + (1 + |Z_0|)|Z_0| + |Z_0|^2
Let M be this number. Now, take delta smaller than eps/M and also
smaller than 1. It follows that
|Z^3 - Z_0^3| < (eps/M)*M = eps.
Best regards,
Jose Carlos Santos
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| | From: | emailzul at starhub.net.sg | | Subject: | Re: Solving complex limit using epsilon delta | | Date: | 23 Jan 2005 08:05:55 -0800 |
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| Hello Jose Carlos Santos,
It seems you have always been helping me in giving me hints to solve
questions on complex.
Are you into complex analysis as well or are you a professor?
Regards
Zul.
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| | From: | José_Carlos_Santos | | Subject: | Re: Solving complex limit using epsilon delta | | Date: | Sun, 23 Jan 2005 17:39:53 +0000 |
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| emailzul@starhub.net.sg wrote:
> It seems you have always been helping me in giving me hints to solve
> questions on complex.
>
> Are you into complex analysis as well or are you a professor?
I was a complex analysis teacher for ten consecutive years.
Best regards,
Jose Carlos Santos
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