1. simple limits question

Please have a look at this question:

Prove that limit, as z tends to a, of (z^2 +c) = a^2 + c

The question is trivial to solve using the theorem that the limit of a polynomial in the set of complex numbers at a is P(a) or even using the theorems about the limits of f(x) + g(x) and limit of f(x)*g(x) where both f(x) and g(x) have limit at a.

However I have to solve it using the epsilon delta definition of limit of complex functions i.e. given an epsilon, show that a delta can be constructed which would confine the mod of [f(x) - (a^2+c)] within the given epsilon. Despite several attempts, I have failed. Please can someone help?

Many Thanks

2. Originally Posted by avicenna
Please have a look at this question:

Prove that limit, as z tends to a, of (z^2 +c) = a^2 + c

The question is trivial to solve using the theorem that the limit of a polynomial in the set of complex numbers at a is P(a) or even using the theorems about the limits of f(x) + g(x) and limit of f(x)*g(x) where both f(x) and g(x) have limit at a.

However I have to solve it using the epsilon delta definition of limit of complex functions i.e. given an epsilon, show that a delta can be constructed which would confine the mod of [f(x) - (a^2+c)] within the given epsilon. Despite several attempts, I have failed. Please can someone help?

Many Thanks
If you had asked this only for real numbers then i could show the proof here. I have not read calculus in complex variable so i don't know.
But in your text book there must be the proof of the fact that "limit of a polynomial in the set of complex numbers at a is P(a)". I guess the text books prove these theorems using epsilon-delta method for complex domain as well. I don't know if this helps but still....

Yes, Chrichill and Brown (the book I am using) does have proofs of the theorems you have mentioned in the set of complex numbers and indeed proving the given statement using those theorems is trivial. But I need to solve the question using the basic epsilon delta notation only, without the using the theorms on limits of polynomials or limits of functions multiplied/added together.

I try to work backwards from what I am trying to show ( i.e. for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon), I use the triangle inequality a few times and arrive at a Delta. When I try to then work forward from that Delta to construct a step-by-step proof, that my Delta would indeed leed to the statement that I am trying to prove, it does not work. No matter how I construct the Delta (I have tried a few different ways), I cannot work forward to the statement:

for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon

Can anyone help, or otherwise convince me that I cannot do without employ the theorems of limits mentioned in the post.

Thanks a lot

4. Originally Posted by avicenna
Prove that limit, as z tends to a, of (z^2 +c) = a^2 + c
Just take note that $|(z^2+c)-(a^2+c)|=|z-a||z+a|.$
If we make sure that $|z-a|<1$ then we know $|z+a|<1+2|a|$.

If $\varepsilon > 0$ choose $\delta = \min \left\{ {1,\frac{\varepsilon }{{1 + 2\left| a \right|}}} \right\}$.

5. Originally Posted by avicenna

Yes, Chrichill and Brown (the book I am using) does have proofs of the theorems you have mentioned in the set of complex numbers and indeed proving the given statement using those theorems is trivial. But I need to solve the question using the basic epsilon delta notation only, without the using the theorms on limits of polynomials or limits of functions multiplied/added together.

I try to work backwards from what I am trying to show ( i.e. for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon), I use the triangle inequality a few times and arrive at a Delta. When I try to then work forward from that Delta to construct a step-by-step proof, that my Delta would indeed leed to the statement that I am trying to prove, it does not work. No matter how I construct the Delta (I have tried a few different ways), I cannot work forward to the statement:

for all z such that 0<mod[z-a]<Delta, mod[(z^2+c)-(a^2+c)] < Epsilon

Can anyone help, or otherwise convince me that I cannot do without employ the theorems of limits mentioned in the post.

Thanks a lot
i checked out the book. in chapter 2 they have discussed the theorems on limits in section 12 of the book.
The have proved it using epsilon-delta method. so you have to just read the proof and wherever they have written f(z) yo should read z^2+c and you will have proved the thing using epsilon delta. do you see what i am trying to say??

6. I know what you mean and that would be easy to do. However, Plato has provided what I was looking for.

7. Hi Plato,

Thanks a lot for your help. I do not understand how you arrived at the third line?

8. ok, got you. Thanks a million Plato.