Question 2.6.3: Using Theorem 2.1 to Compute a Limit. Use Theorem 2.1 to com......

Since f(z) = z² +i = x² −y² +(2xy + 1) i, we can apply Theorem 2.1 with u(x, y) = x² − y², v(x, y) = 2xy + 1, and z_{0} = 1+i. Identifying x_{0} = 1 and y_{0} =1, we find u_{0} and v_{0} by computing the two real limits:

Since both of these limits involve only multivariable polynomial functions, we can use (13) to obtain:

\underset{(x,y)\rightarrow (x_{0},y_{0})}{lim}p(x,y)=p(x_{0},y_{0})     (13)

and              v_{0}= \underset{(x,y)\rightarrow (1,1)}{lim}(2xy+1)=2\cdot 1 \cdot 1+1=3,

and so L  =  u_{0}  +   iv_{0}  =  0  +  i(3)  =  3i Therefore, \underset{z\rightarrow 1+i}{lim}(z^{2}+i) =3i.