Calculating Limits of Complex Functions

[Mattofix]

Homework Statement

Compute limit as n-> infinity of

a)zn = exp(in^2)/(1+in^2)

b)zn = 1/(n + i)

Homework Equations

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The Attempt at a Solution

These are 2 examples of a series of questions i have to complete. I can see that i need to calculate the limit for a complex function but i have not come across this in lectures yet. Could you please point me in the right direction, maybe i need to use a certain rule/law/method?

 

[CompuChip]

Hint: look at the moduli first, that is, consider
$$\lim_{n \to \infty} |z_n|$$

 

[HallsofIvy]

Finding the limit in the complex numbers is pretty much the same as real numbers. As compuchip said, you need to look at $|z_n|= \sqrt{(x_n)^2+ (y_n)^2}$. Although it won't give you the limit exactly, you should try calculating a few terms of the sequence to get an idea of what's happening. Have you done that?

 

[Mattofix]

should i be using the subtitution of exp(ia) = cos a - isin a ? or is there a better method?

 

[Mattofix]

yeah - i know that it tends to 0 - but I am not too sure about what i use for the xn and yn parts and what that eventually gives me - if i can use exp(ia) = cos a - isin a i think i have crcked it?

 

[CompuChip]

You can always write it out in real and imaginary parts and notice that
$$\lim_{z \to 0} f(z) = \left( \lim_{z \to 0} \operatorname{Re}(f(z)) \right) + \mathrm{i} \left( \lim_{z \to 0} \operatorname{Im}(f(z)) \right)$$.
Then you can indeed use exp(i a) = cos(a) + i sin(a) for the first one, and use that the real and imaginary parts of a fraction can be determined by writing
$$\frac{\alpha}{z} = \frac{\alpha}{z} \frac{\bar z}{\bar z}$$
with $\bar z$ the complex conjugate of z.

So if you think you have a), can you show us your work?