Question 2

Compute the following limits, or explain why the limit in question does not exist.

• $\mathcal{A} = \displaystyle{\lim_{z \rightarrow 1 - i} \frac{(z + 5i)(z^2 + 3z - 3 + 5i) }{z^2 + 2i}}$
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
• $\mathcal{B} = \displaystyle{\lim_{z \rightarrow 0} \frac{(\Re\left(z^2\right) - \vert z\vert^2)}{\Im\left(\overline{z}^2\right)}}$.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
• $\mathcal{C} = \displaystyle{\lim_{z \rightarrow 0} \left( \frac{(1 - 2z)^{-2} - 1 - 4z}{\sin^2(2z)}\right)}$.
  
  Use series methods for the limit $\mathcal{C}$.