### Calculus - Evaluating Limits **Problem Statement:** Evaluate the following limit: \[ \lim_{{x \to 8}} \frac{{3 \cos x}}{{7x}} \] **Explanation:** To find the value of this limit, we can follow these steps: 1. Substitute \( x = 8 \) into the function: \[ \frac{{3 \cos 8}}{{7 \cdot 8}} \] 2. Calculate the cosine of 8 (note: this step may require a calculator to get an accurate decimal value): \[ \cos 8 \] (in radians, \(\cos 8\) generally needs a calculator for precise value) 3. Multiply the result by 3 and divide by \(7 \cdot 8\): \[ \frac{{3 \cdot \cos 8}}{{56}} \] By performing these operations, you'll get the evaluated limit. **Note:** If any simplification or additional limit properties are required, further algebraic manipulation or L'Hopital's rule (for indeterminate forms) might be typically used. This approach demonstrates the direct substitution method for evaluating limits as \( x \) approaches a specific value. --- This transcription and explanation are intended for educational purposes to assist in understanding how to evaluate such limits.
### Calculus - Evaluating Limits **Problem Statement:** Evaluate the following limit: \[ \lim_{{x \to 8}} \frac{{3 \cos x}}{{7x}} \] **Explanation:** To find the value of this limit, we can follow these steps: 1. Substitute \( x = 8 \) into the function: \[ \frac{{3 \cos 8}}{{7 \cdot 8}} \] 2. Calculate the cosine of 8 (note: this step may require a calculator to get an accurate decimal value): \[ \cos 8 \] (in radians, \(\cos 8\) generally needs a calculator for precise value) 3. Multiply the result by 3 and divide by \(7 \cdot 8\): \[ \frac{{3 \cdot \cos 8}}{{56}} \] By performing these operations, you'll get the evaluated limit. **Note:** If any simplification or additional limit properties are required, further algebraic manipulation or L'Hopital's rule (for indeterminate forms) might be typically used. This approach demonstrates the direct substitution method for evaluating limits as \( x \) approaches a specific value. --- This transcription and explanation are intended for educational purposes to assist in understanding how to evaluate such limits.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Calculus - Evaluating Limits
**Problem Statement:**
Evaluate the following limit:
\[ \lim_{{x \to 8}} \frac{{3 \cos x}}{{7x}} \]
**Explanation:**
To find the value of this limit, we can follow these steps:
1. Substitute \( x = 8 \) into the function:
\[ \frac{{3 \cos 8}}{{7 \cdot 8}} \]
2. Calculate the cosine of 8 (note: this step may require a calculator to get an accurate decimal value):
\[ \cos 8 \] (in radians, \(\cos 8\) generally needs a calculator for precise value)
3. Multiply the result by 3 and divide by \(7 \cdot 8\):
\[ \frac{{3 \cdot \cos 8}}{{56}} \]
By performing these operations, you'll get the evaluated limit.
**Note:**
If any simplification or additional limit properties are required, further algebraic manipulation or L'Hopital's rule (for indeterminate forms) might be typically used.
This approach demonstrates the direct substitution method for evaluating limits as \( x \) approaches a specific value.
---
This transcription and explanation are intended for educational purposes to assist in understanding how to evaluate such limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7a11c6-35df-4daf-855c-043e8f69fb75%2F6cb086ca-5ed6-4769-985c-9e4982282c3e%2F58bauv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus - Evaluating Limits
**Problem Statement:**
Evaluate the following limit:
\[ \lim_{{x \to 8}} \frac{{3 \cos x}}{{7x}} \]
**Explanation:**
To find the value of this limit, we can follow these steps:
1. Substitute \( x = 8 \) into the function:
\[ \frac{{3 \cos 8}}{{7 \cdot 8}} \]
2. Calculate the cosine of 8 (note: this step may require a calculator to get an accurate decimal value):
\[ \cos 8 \] (in radians, \(\cos 8\) generally needs a calculator for precise value)
3. Multiply the result by 3 and divide by \(7 \cdot 8\):
\[ \frac{{3 \cdot \cos 8}}{{56}} \]
By performing these operations, you'll get the evaluated limit.
**Note:**
If any simplification or additional limit properties are required, further algebraic manipulation or L'Hopital's rule (for indeterminate forms) might be typically used.
This approach demonstrates the direct substitution method for evaluating limits as \( x \) approaches a specific value.
---
This transcription and explanation are intended for educational purposes to assist in understanding how to evaluate such limits.
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