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Complex limit (again)

  1. May 14, 2015 #1

    jjr

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    1. The problem statement, all variables and given/known data
    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##

    2. Relevant equations


    3. 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
     
  2. jcsd
  3. May 14, 2015 #2
    edit: mistake
     
    Last edited: May 14, 2015
  4. May 14, 2015 #3

    jjr

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    I will take your word, because it makes perfect sense and it's the correct answer (suppose I should've mentioned that). Thanks.
     
  5. May 14, 2015 #4
    No there was a mistake there. The first term should be multiplied by ##\sqrt{z} + i##

    You get $$\frac{(\sqrt{z}-i)(\sqrt{z}+i)}{\sqrt{(z-1)(z+1)}(\sqrt{z} + i)} + \frac{\sqrt{z+1}}{\sqrt{(z-1)(z+1)}} = \frac{\sqrt{z+1}}{(\sqrt{z-1})(\sqrt{z} + i )} + \frac{1}{\sqrt{z-1}}$$
     
    Last edited: May 14, 2015
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