- #1
jjr
- 51
- 1
Homework Statement
Calculate the following limit if it exists
## \lim_{z\to -1}\frac{\sqrt{z}-i+\sqrt{z+1}}{\sqrt{z^2-1}} ##
the branch of root is chosen so that ## \sqrt{-1}=i##
Homework Equations
The Attempt at a Solution
I tried most of the same things that I tried earlier today ( https://www.physicsforums.com/threads/complex-limit-help.813800/ ).
1. No obvious way to simplify the expression and get rid of the zero in the denominator.
2. Tried to multiply the numerator and denominator with the complex conjugate ## \sqrt{\bar{z}^2-1} ##, which did give me a ## \sqrt{2} ## in the denominator. The problem with this is that if ## z ## goes to ##-1##, then I suppose ##\bar{z}## goes to ## -1 ## as well. This means, in effect, that I multiplied with 0, and I don't think it's legit. (Yields a wrong answer (0) anyway).
3. Tried using l'Hopitals rule, but it doesn't remove any of the radicals which will still go to 0 no matter how many times I apply it.
4. Tried the transformation ## z = re^{i\theta} ##, but can't see how it would get me any further.
Any ideas?
Thanks,
J