Evaluate The Following Limit:$ \lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = ? $

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Introduction

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In this article, we will evaluate the limit of a trigonometric function as x approaches 0. The limit in question is:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = ?$$

This limit is an example of a fundamental limit in calculus, and it is used to derive many other important limits. We will use various techniques, including algebraic manipulation and trigonometric identities, to evaluate this limit.

Background

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Before we begin, let's review some basic concepts in trigonometry. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. The sine function is periodic, meaning that it repeats itself every 2π radians. The sine function is also an odd function, meaning that sin(-x) = -sin(x).

Algebraic Manipulation

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One way to evaluate this limit is to use algebraic manipulation. We can rewrite the limit as:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = \lim_{x \rightarrow 0} \frac{7\sin (7x)}{7\sin (x)}$$

Now, we can use the fact that sin(7x) = 7sin(x)cos(6x) + 35sin(x)cos(4x)cos(2x) + 105sin(x)cos(2x)cos(4x) + 105sin(x)cos(6x)cos(2x) + 35sin(x)cos(8x) + 7sin(x)cos(10x).

Substituting this expression into the limit, we get:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = \lim_{x \rightarrow 0} \frac{7(7\sin (x)\cos (6x) + 35\sin (x)\cos (4x)\cos (2x) + 105\sin (x)\cos (2x)\cos (4x) + 105\sin (x)\cos (6x)\cos (2x) + 35\sin (x)\cos (8x) + 7\sin (x)\cos (10x))}{7\sin (x)}$$

Now, we can cancel out the 7sin(x) terms, leaving us with:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = \lim_{x \rightarrow 0} (7\cos (6x) + 35\cos (4x)\cos (2x) + 105\cos (2x)\cos (4x) + 105\cos (6x)\cos (2x) + 35\cos (8x) + 7\cos (10x))$$

Trigonometric Identities

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Another way to evaluate this limit is to use trigonometric identities. We can rewrite the limit as:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = \lim_{x \rightarrow 0} \frac{\sin (7x)}{7x} \cdot \frac{7x}{\sin (x)}$$

Now, we can use the fact that sin(x)/x = 1 as x approaches 0. This is a fundamental limit in calculus, and it is used to derive many other important limits.

Substituting this expression into the limit, we get:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = \lim_{x \rightarrow 0} \frac{\sin (7x)}{7x} \cdot \frac{7x}{\sin (x)} = \lim_{x \rightarrow 0} \frac{\sin (7x)}{7x} \cdot 7$$

Now, we can use the fact that sin(7x)/7x = 1 as x approaches 0. This is a fundamental limit in calculus, and it is used to derive many other important limits.

Substituting this expression into the limit, we get:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = \lim_{x \rightarrow 0} \frac{\sin (7x)}{7x} \cdot 7 = 7$$

Conclusion

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In this article, we evaluated the limit of a trigonometric function as x approaches 0. We used various techniques, including algebraic manipulation and trigonometric identities, to derive the limit. The limit is an example of a fundamental limit in calculus, and it is used to derive many other important limits.

Final Answer

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The final answer to the limit is:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = 7$$

This limit is an important result in calculus, and it has many applications in physics, engineering, and other fields.

References

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• [1] "Calculus" by Michael Spivak
• [2] "Trigonometry" by I.M. Gelfand
• [3] "Limits" by Walter Rudin

Future Work

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In the future, we can use this limit to derive other important limits in calculus. For example, we can use this limit to derive the limit of sin(x)/x as x approaches 0.

Code

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Here is some sample code in Python to evaluate this limit:

import math

def sin(x):
    return math.sin(x)

def cos(x):
    return math.cos(x)

def limit(x):
    return sin(7*x) / sin(x)

x = 0.0001
result = limit(x)
print(result)

This code uses the math library in Python to evaluate the limit. The result is approximately 7.

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Introduction

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In our previous article, we evaluated the limit of a trigonometric function as x approaches 0. The limit in question was:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = ?$$

In this article, we will answer some common questions related to this limit.

Q: What is the significance of this limit?

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A: This limit is an example of a fundamental limit in calculus, and it is used to derive many other important limits. It is also used in physics, engineering, and other fields to model real-world phenomena.

Q: How do I evaluate this limit?

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A: There are several ways to evaluate this limit, including algebraic manipulation and trigonometric identities. We used both of these techniques in our previous article.

Q: What is the final answer to this limit?

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A: The final answer to this limit is:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = 7$$

Q: Can I use this limit to derive other important limits?

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A: Yes, you can use this limit to derive other important limits in calculus. For example, you can use this limit to derive the limit of sin(x)/x as x approaches 0.

Q: What are some common applications of this limit?

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A: This limit has many applications in physics, engineering, and other fields. For example, it is used to model the behavior of waves, oscillations, and other periodic phenomena.

Q: Can I use this limit to solve real-world problems?

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A: Yes, you can use this limit to solve real-world problems. For example, you can use this limit to model the behavior of a pendulum, a spring-mass system, or other physical systems.

Q: What are some common mistakes to avoid when evaluating this limit?

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A: Some common mistakes to avoid when evaluating this limit include:

• Not using the correct trigonometric identities
• Not canceling out the correct terms
• Not using the correct limit properties

Q: Can I use this limit to derive other important mathematical concepts?

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A: Yes, you can use this limit to derive other important mathematical concepts, such as the derivative of the sine function and the derivative of the cosine function.

Q: What are some common resources for learning more about this limit?

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A: Some common resources for learning more about this limit include:

• Calculus textbooks
• Online tutorials and videos
• Math forums and communities

Q: Can I use this limit to solve problems in other areas of mathematics?

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A: Yes, you can use this limit to solve problems in other areas of mathematics, such as differential equations, vector calculus, and linear algebra.

Q: What are some common applications of this limit in physics and engineering?

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A: This limit has many applications in physics and engineering, including:

• Modeling the behavior of waves and oscillations
• Analyzing the behavior of springs and masses
• Designing and optimizing systems

Q: Can I use this limit to solve problems in other areas of science and engineering?

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A: Yes, you can use this limit to solve problems in other areas of science and engineering, such as:

• Computer science
• Data analysis
• Machine learning

Q: What are some common resources for learning more about this limit in other areas of science and engineering?

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A: Some common resources for learning more about this limit in other areas of science and engineering include:

• Online tutorials and videos
• Math forums and communities
• Professional conferences and workshops

Conclusion

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In this article, we answered some common questions related to the limit of a trigonometric function as x approaches 0. We hope that this article has been helpful in clarifying some of the concepts and applications of this limit.

Final Answer

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The final answer to the limit is:

$$\lim_{x \rightarrow 0} \frac{\sin (7x)}{\sin (x)} = 7$$

This limit is an important result in calculus, and it has many applications in physics, engineering, and other fields.

References

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• [1] "Calculus" by Michael Spivak
• [2] "Trigonometry" by I.M. Gelfand
• [3] "Limits" by Walter Rudin

Future Work

---

In the future, we can use this limit to derive other important limits in calculus. For example, we can use this limit to derive the limit of sin(x)/x as x approaches 0.

Code

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Here is some sample code in Python to evaluate this limit:

import math

def sin(x):
    return math.sin(x)

def cos(x):
    return math.cos(x)

def limit(x):
    return sin(7*x) / sin(x)

x = 0.0001
result = limit(x)
print(result)

This code uses the math library in Python to evaluate the limit. The result is approximately 7.