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Complex Limit

  1. Sep 12, 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 to that ##\sqrt{-1}=i##

    2. Relevant equations
    3. The attempt at a solution

    By inserting ##z=-1## directly, I get a ##\frac{0}{0}## expression.

    I don't see how the expression can be simplified before inserting the limit, that seems out of the question.
    I have a few theorems involving points at infinity, but none of them apply here.
    I considered making the denominator real by multiplying by its complex conjugate, but the numerator will still be 0 when multiplied by any number.

    The answer is supposed to be ##1/(i\sqrt{2})##.

    Not looking for a solution or quick fix, would be very happy if someone could help me the first step of the way.

    Sincerely,
    J
     
  2. jcsd
  3. Sep 12, 2015 #2
    Hint: Try multiplying the expression by: ##\frac { \sqrt { z + i } }{ \sqrt { z + i } } ##

    Moderator note: I revised the expression above to convey what FaroukYasser meant, but did not write.
     
    Last edited by a moderator: Sep 12, 2015
  4. Sep 12, 2015 #3

    jjr

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    Thank you, got it now!

    ## \lim_{i\to -1} \frac{\sqrt{z}-1+\sqrt{z+1}}{\sqrt{z^2-1}} ##

    ##\to \lim_{i\to -1} \frac{\sqrt{z}-1+\sqrt{z+1}}{\sqrt{(z+1)(z-1)}} ##

    ##\to \lim_{i\to -1} \frac{1}{\sqrt{z-1}} = \frac{1}{i\sqrt{2}} ##
     
  5. Sep 12, 2015 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    Is your numerator equal to ##\sqrt{z} \: - i \: + \sqrt{z+1}##, or is it ##\sqrt{z-i} + \sqrt{z+1}## (or maybe even ##\sqrt{z-i} + \sqrt{z+i}## )? You wrote the first.
     
  6. Sep 12, 2015 #5

    jjr

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    The first
     
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