Complex Analysis - Maths KU

2.6 Limits and Continuity 129
In Problems 9–16, use Theorem 2.2 and the basic limits (15) and (16) to compute
the given complex limit.
9.
� � 2
lim z − z
z→2−i
� �
5 2
10. lim z − z + z
z→i
11. lim
z→eiπ/4 �
z + 1
�
z
12. lim
z→1+i
z 2 +1
z2 − 1
13.
z
lim
z→−i
4 − 1
z + i
14. lim
z→2+i
z 2 − (2 + i) 2
z − (2 + i)
(az + b) − (az0 + b)
15. lim
z→z0 z − z0
Re(z)
17. Consider the limit lim
z→0 Im(z) .
16. lim
z→−3+i √ 2
z +3− i √ 2
z 2 +6z +11
(a) What value does the limit approach as z approaches 0 along the line y = x?
(b) What value does the limit approach as z approaches 0 along the imaginary
axis?
Re(z)
(c) Based on your answers for (a) and (b), what can you say about lim
z→0 Im(z) ?
18. Consider the limit lim
z→i (|z| + iArg (iz)).
(a) What value does the limit approach as z approaches i along the unit circle
|z| = 1 in the first quadrant?
(b) What value does the limit approach as z approaches i along the unit circle
|z| = 1 in the second quadrant?
(c) Based on your answers for (a) and (b), what can you say about
lim (|z| + iArg (iz))?
z→i
�
z
�2 19. Consider the limit lim .
z→0 ¯z
(a) What value does the limit approach as z approaches 0 along the real axis?
(b) What value does the limit approach as z approaches 0 along the imaginary
axis?
�
z
�2 (c) Do the answers from (a) and (b) imply that lim exists? Explain.
z→0 ¯z
(d) What value does the limit approach as z approaches 0 along the line y = x?
�
z
�2 (e) What can you say about lim ?
z→0 ¯z
x2 − x2 − y 2
y2 �
i .
� 2
2y
20. Consider the limit lim
z→0
(a) What value does the limit approach as z approaches 0 along the line y = x?
(b) What value does the limit approach as z approaches 0 along the line
y = −x?
� 2
2y
(c) Do the answers from (a) and (b) imply that lim
z→0 x2 − x2 − y 2
y2 �
i exists?
Explain.