Birla Institute of Technology and Science, Pilani-K K Birla Goa Campus
Second Semester 2022-2013
MATHEMATICS-II (MATH F112)
Tutorial Sheet-7
Topic: Functions and their Limits and Continuity
1. Sketch the following sets and determine which are domains, which are neither open
nor closed, which are bounded:
(a) |z − 2 + i| ≤ 1 (b) |2z + 3| > 4 (c) ℑ(z) > 1 (d) ℑ(z) = 1.
Here ℑ(z) is the imaginary part of z it is also denoted by Imz.
2. In each case, sketch the closure of the set:
(a) −π < argz < π, (z ̸= 0) (b) |ℜ(z)| < |z| (c) ℜ z1 ≤ 12 (d) ℜ(z 2 ) > 0.
Here ℜ(z) is the real part of z it is also denoted by Rez.
3. Show S be the open set consisting of all points z such that |z| < 1 or |z − 1| < 1. State
why S is not connected.
4. Determine the accumulation points of each of the following sets:
(a) zn = in (n = 1, 2, . . .)
(c) zn = in /n (n = 1, 2, . . .)
(b) 0 ≤ arg z < π/2 (z ̸= 0)
(d) zn = (−1)n (1 + i) n−1
(n = 1, 2, . . .)
n
5. Show that ||z| − |w|| ≤ |z − w|, (z, w ∈ C)
6. Draw the subsets of the complex plane and also determine which subsets are open,
closed or connected.
(a) {z : |z| < 1}.
(d) {z : ez = 1}.
(b) {z : Re z = 1}.
(e) {z : |z − 1| < |z + 1|}.
(c) {z : Re (z 2 ) < 1}.
(f) {z : 0 < |z − 1| < 2}.
7. Let z1 , z2 be in the disc {z : |z| ≤ 1} such that |z1 − z2 | ≥ 1 then show that |z1 + z2 | ≤
√
3.
8. Let f : C \ {1} → C, be defined as f (z) = z+1
, shade the image of the following sets
z−1
(a) {z : |z| = 2}.
(c) {z : z 2 = 4}.
(b) {z = x + iy : 2x + 3y = 1}.
(d) {z : 0 < |z| < 21 }.
1
9. For each of the function below, describe the domain of the definition that is understood:
1
2
z +1
z
(b) f (z) =
z + z̄
(a) f (z) =
(c) f (z) = Arg
(d) f (z) =
1
z
1
1 − |z|2
10. Write the function f (z) = z 3 + z + 1 in the form f (z) = u(x, y) + iv(x, y).
11. Suppose that f (z) = x2 − y 2 − 2y + i(2x − 2xy), where z = x + iy. Use the expression
x=
z + z̄
2
and y =
z − z̄
2i
to write f (z) in terms of z, and simplify the result.
12. Write the function
f (z) = z +
1
(z ̸= 0)
z
in the form f (z) = u(r, θ) + iv(r, θ).
13. Use ε − δ definition of limit to prove that
(a) lim Re z = Re z0 ;
z2
= 0.
z→0 z
(b) lim z = z0 ;
z→z0
(c) lim
z→z0
14. Let a, b and c denote complex constants. Then use ε − δ definition of limit to show
that
(a) lim (az + b) = az0 + b;
(c) lim [x + i(2x + y)] = 1 + i, where
z→z0
(b) lim (z
z→z0
2
z→1−i
(z = x + iy).
+ c) = z02 + c;
15. Show that the limit of the function f (z) =
z 2
z̄
as z tends to 0 does not exist.
16. Use the theorem in Sec. 17 from text book to show that
1
= ∞;
z→1 (z − 1)3
4z 2
(a) lim
= 4;
z→∞ (z − 1)2
(b) lim
z2 + 1
(c) lim
= ∞.
z→∞ z − 1
17. With the aid of the theorem in Sec. 17 from text book, show that when
T (z) =
az + b
cz + d
(ad − bc ̸= 0),
(a) lim T (z) = ∞ if c = 0;
z→∞
(b) lim T (z) =
z→∞
a
and lim T (z) = ∞ if c ̸= 0.
z→d/c
c
18. Find which of the following limits exists:
2
1−z
.
z→1 1 − z
(a) lim
z2 − z2
.
z→0
z
(b) lim
(c) lim
z
z→0 Rez
.
19. Determine whether each of the following statements is true or false. Justify your answer
with a proof or a counterexample.
Re z 2
Im z 2
and
lim
does not exist.
z→0 |z|2
z→0 |z|2
Re (z) + Im (z)
, lim f (z) does not exist.
(b) For f (z) =
z→0
|z|2
(a) Each of the limits lim
Re (z)
is continuous for all z with Im z ̸= 0 and discontinIm (z)
uous for all z with Im z = 0. Also, no discontinuity of f (z) is removable.
(c) The function f (z) =
(d) The function f (z) = (2 + z)Arg z is continuous on the punctured plane C \ {0}.
(e) The function f (z) = (Arg z)2 is continuous on the punctured plane C \ {0}.
20. Discuss the continuity of the following complex-valued functions at z = 0:

 (Re z)(Im z)
if z ̸= 0
|z|2
(a) f (z) =
.

0
if z = 0

 Im z
if z ̸= 0
|z|
(b) f (z) =
.

0
if z = 0
(c) f (z) =
Re z
for z ∈ C.
1 + |z|
(d) f (z) = |Re(z)Im(z)| for z ∈ C.

 sin |z|
if z ̸= 0
|z|
(e) f (z) =
.

1
if z = 0

2

 1 − exp(−|z| )
if z ̸= 0
|z|2
.
(f) f (z) =

1
if z = 0
3