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The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.?
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The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only l...
Answer:
To determine the limit points of the set {1/n sin(1/n): n ∈ N}, we need to find the values that the set approaches as n approaches infinity.
1. Evaluating the expression:
Let's evaluate the expression for a few values of n to understand the behavior of the set:
When n = 1: 1/1 sin(1/1) = 1 sin(1) ≈ 0.8415
When n = 2: 1/2 sin(1/2) = 1/2 sin(1/2) ≈ 0.4207
When n = 3: 1/3 sin(1/3) = 1/3 sin(1/3) ≈ 0.3473
...
2. Looking for patterns:
As n approaches infinity, we observe that the values of 1/n sin(1/n) are getting closer and closer to 0. However, there are also oscillations present in the values. To determine the limit points, we need to investigate the behavior of the set further.
3. Analyzing the oscillations:
Let's consider the function f(x) = sin(x)/x. As x approaches 0, f(x) approaches 1. This means that the values of sin(1/n)/(1/n) approach 1 as n approaches infinity. However, the additional factor of 1/n in the set {1/n sin(1/n): n ∈ N} causes the values to approach 0.
4. Identifying the limit points:
From the analysis above, we can conclude that the set {1/n sin(1/n): n ∈ N} has only one limit point, which is 0. The oscillations caused by the sin(1/n) term prevent the set from having any other limit points.
Therefore, the correct answer is (b) Only limit point its 0.
To determine the limit points of the set {1/n sin(1/n): n ∈ N}, we need to find the values that the set approaches as n approaches infinity.
1. Evaluating the expression:
Let's evaluate the expression for a few values of n to understand the behavior of the set:
When n = 1: 1/1 sin(1/1) = 1 sin(1) ≈ 0.8415
When n = 2: 1/2 sin(1/2) = 1/2 sin(1/2) ≈ 0.4207
When n = 3: 1/3 sin(1/3) = 1/3 sin(1/3) ≈ 0.3473
...
2. Looking for patterns:
As n approaches infinity, we observe that the values of 1/n sin(1/n) are getting closer and closer to 0. However, there are also oscillations present in the values. To determine the limit points, we need to investigate the behavior of the set further.
3. Analyzing the oscillations:
Let's consider the function f(x) = sin(x)/x. As x approaches 0, f(x) approaches 1. This means that the values of sin(1/n)/(1/n) approach 1 as n approaches infinity. However, the additional factor of 1/n in the set {1/n sin(1/n): n ∈ N} causes the values to approach 0.
4. Identifying the limit points:
From the analysis above, we can conclude that the set {1/n sin(1/n): n ∈ N} has only one limit point, which is 0. The oscillations caused by the sin(1/n) term prevent the set from having any other limit points.
Therefore, the correct answer is (b) Only limit point its 0.
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The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.?
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The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.?.
The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The set { 1/n Sin 1/ n: nEN } has a . only limit point its 1 b. only limit point its o C. only limit point its-1 d. there are three limit Point it 0,-1 , 1.?.
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