11/28/22, 2:37 AM
Complex Analysis Objective Questions and Answers - Sanfoundry
11/28/22, 2:37 AM
limh→0
limh→0
Complex Analysis Objective Questions and Answers - Sanfoundry
f (1+h)−f (1)
h
(1+h+5)−(1+5)
h
=1
Since the two limits are not equal, the function is not differentiable.
Complex Function Theory Questions and
Answers – Differentiability
3. Which of the following is correct about the function f(x)=|x2+18x+81|?
a) Left and right limits are equal and hence it is differentiable
b) Limits are not important to determine differentiability
c) Left and right limits are equal and hence it is not differentiable
d) Left and right limits are not equal and hence it is not differentiable
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This set of Complex Analysis Objective Questions & Answers focuses on “Differentiability”.
View Answer
1. Which of the following is true?
a) Differentiability does not imply continuity
b) Differentiability implies continuity
c) Continuity implies differentiability
d) There is no relation between continuity and differentiable
Answer: d
Explanation: f(x)=|(x+9)2|
Since the break is at x=-9, we frame for the function for both the sides of -9.
f(x)=(x+9)2, if x>-9 and f(x)=(x+9)2, if x≤-9
We know that, a function is not differentiable at point x=a, if either lim
not exist or is infinity. We check limits for both the cases of the function.
Left limit: Here, a=-10.
h→0
View Answer
Answer: b
Explanation: Any function that is differentiable is definitely continuous. If a function is not
continuous, it cannot be differentiable either. However, continuity cannot always imply
differentiability. Sometimes, functions that are continuous are not differentiable.
limh→0
limh→0
2
h
2
1 +h −2h−1
2
h
f (1+h)−f (1)
h
2
limh→0
2
(10+h+9) −(10+9)
h
2
limh→0
2
2
10 +h +20h−10
h
=20
Since the two limits are not equal, the function is not differentiable.
View Answer
Answer: d
Explanation: f(x)=(x+5), if x>-5 and f(x)=-(x+5), if x≤-5
Since the break is at x=-5, we calculate the limit at this point.
We know that, a function is not differentiable at point x=a, if either lim
not exist or is infinity. We check limits for both the cases of the function.
Left limit: Here, a=-5.
h→0
limh→0
h
=-2
Right limit: Here, a=10.
2. Which of the following is correct about the function f(x)=|x+5|?
a) Left and right limits are equal and hence it is differentiable
b) Left and right limits are not equal and hence it is differentiable
c) Left and right limits are equal and hence it is not differentiable
d) Left and right limits are not equal and hence it is not differentiable
limh→0
does
(−10+h+9) −(−10+9)
2
limh→0
h
f (−10+h)−f (−10)
2
limh→0
f (a+h)−f (a)
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f (a+h)−f (a)
h
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does
f (−5+h)−f (−5)
h
−(−5+h+5)−(−(−5+5))
h
=-1
Right limit: Here, a=1.
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Complex Analysis Objective Questions and Answers - Sanfoundry
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Complex Analysis Objective Questions and Answers - Sanfoundry
In general the limits are discussed at origin, if nothing is specified.
f(x, y)=x+iy+ix+i2y
f(x, y)=(x-y)+i(x+y)
Left limit: lim
(x-y)+i(x+y)
x→0
limy→0
-y+iy
=0
Right limit: lim
4. Which of the following is true about f(z)=z2?
a) Continuous and differentiable
b) Continuous but not differentiable
c) Neither continuous nor differentiable
d) Differentiable but not continuous
limx→0
f’(z)=lim
limδz→0
x→0
f (z+δz)−f (z)
δz→0
δz
2
δz
should exist.
View Answer
2
(z+δz) −(z)
δz
Answer: a
Explanation: z=x+iy
In general the limits are discussed at origin, if nothing is specified.
f(x, y)=(x+iy)2+2(x+iy)
f(x, y)=x2-y2+2xiy+2x+2iy
Left limit: lim
x2-y2+2xiy+2x+2iy
2
limδz→0
δz
δz+iδz
6. Which of the following is true about f(z)=z2+2z?
a) Continuous and differentiable
b) Continuous but not differentiable
c) Neither continuous nor differentiable
d) Differentiable but not continuous
(x+iy)2
y→0
=0
Both the limits are equal, therefore the function is continuous. To check differentiability,
limδz→0
should exist.
y→0
=0
Right limit: lim
f’(z)=lim
δz
z+δz+iz+iδz−z+iz
Check this: Engineering Mathematics Books | Probability and Statistics MCQ
x→0
x2
f (z+δz)−f (z)
δz→0
=z+i
Since f’(z) exists, the function is differentiable as well.
Answer: a
Explanation: z=x+iy
In general the limits are discussed at origin, if nothing is specified.
f(x,y)=(x+iy)2
Left limit: lim
(x+iy)2
limx→0
x→0
x+ix
limδz→0
i2y2
x+iy+ix+i2y
y→0
=0
Both the limits are equal, therefore the function is continuous. To check differentiability,
View Answer
limy→0
y→0
2z(δz)+(δz)
δz
=lim
(2z+(δz))
=2z
Since f’(z) exists, the function is differentiable as well.
δz→0
x→0
5. Which of the following is true about f(z)=z+iz?
a) Continuous and differentiable
b) Continuous but not differentiable
c) Neither continuous nor differentiable
d) Differentiable but not continuous
y→0
limy→0
=0
Right limit: lim
limx→0
View Answer
x2+2x
x2-y2+2xiy+2x+2iy
y→0
x→0
=0
Both the limits are equal, therefore the function is continuous. To check differentiability,
Answer: a
Explanation: z=x+iy
https://www.sanfoundry.com/complex-analysis-objective-questions-answers/
-y2+2iy
f’(z)=lim
3/9
f (z+δz)−f (z)
δz→0
δz
should exist.
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Complex Analysis Objective Questions and Answers - Sanfoundry
2
limδz→0
limδz→0
(z+δz) +2(z+δz)−(z
2
11/28/22, 2:37 AM
+2z)
8. Which of the following is true about f(z)=
a) Continuous and differentiable
b) Continuous but not differentiable
c) Neither continuous nor differentiable
d) Differentiable but not continuous
δz
z
2
2
+(δz) +2z(δz)+2z+2(δz)−z
2
−2z
δz
=2z+2
Since f’(z) exists, the function is differentiable as well.
7. Which of the following is true about f(z)=
a) Continuous and differentiable
b) Continuous but not differentiable
c) Neither continuous nor differentiable
d) Differentiable but not continuous
z+iz
z
2
Complex Analysis Objective Questions and Answers - Sanfoundry
2
(x+iy) +(ix+i y)
f(x, y)=
(x+iy) +(ix−y)
2
2
2
(x+iy)
Left limit: lim
2
limy→0
(x+iy)
2
(x+iy)
y→0
limy→0
limy→0
i(x+y)+(x−y)
=
=0
2
(x+iy) +(ix−y)
x→0
2
(x+iy)
2
(iy) +(−y)
2
(iy)
2
2
x→0
?
(x+iy)
2
Left limit: lim
2
2
y→0
f(x, y)=
z
2
2
f(x, y)=
2
i(x+y)+(x−y)
2
+(iz)
Answer: a
Explanation: z=x+iy
In general the limits are discussed at origin, if nothing is specified.
Answer: c
Explanation: z=x+iy
In general the limits are discussed at origin, if nothing is specified.
x+iy+ix+i y
2
View Answer
?
View Answer
f(x, y)=
z
2
−(y) +(y)
2
−(y)
−1+1
2
(x+iy)
i(y)+(−y)
−1
2
(iy)
Right limit: lim
=does not exist
Since, the left limit itself does not exist, the function is not continuous. If a function is not
continuous, it cannot be differentiable as well.
2
2
(x+iy) +(ix−y)
y→0
2
(x+iy)
x→0
2
limx→0
2
(x) +(ix)
2
(x)
2
limy→0
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2
(x) −(x)
2
(x)
=
=0
Both the limits are equal, therefore the function is continuous. To check differentiability,
1−1
1
f(z)=
z
2
−z
z
2
f’(z)=lim
limδz→0
2
=0
f (z+δz)−f (z)
δz→0
δz
should exist.
0−0
δz
=0
Since f’(z) exists, the function is differentiable as well.
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Complex Analysis Objective Questions and Answers - Sanfoundry
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Complex Analysis Objective Questions and Answers - Sanfoundry
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