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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compute the limit
![### Understanding Limits Involving Trigonometric and Logarithmic Functions
In this section, we explore the behavior of a specific mathematical expression as the variable approaches a particular value.
Consider the following limit:
\[ \lim_{{x \to 0^{+}}} (\sin x)^{\frac{2}{\ln x}} \]
#### Components of the Limit Expression:
1. **\( \sin x \)**: This denotes the sine function, which is a standard trigonometric function.
2. **\( \ln x \)**: This denotes the natural logarithm of \( x \).
3. **\( x \to 0^{+} \)**: This indicates that \( x \) approaches 0 from the positive side (i.e., \( x \) remains positive).
4. **Exponent \( \frac{2}{\ln x} \)**: This part of the expression involves a division of 2 by the natural logarithm of \( x \), which becomes particularly interesting as \( x \) approaches 0 from the positive side.
#### Explanation of the Limit:
The expression analyses the behavior of \( (\sin x)^{\frac{2}{\ln x}} \) as \( x \) gets closer to 0 from the right-hand side.
- The sine function, \( \sin x \), oscillates between -1 and 1.
- The natural logarithm \( \ln x \) tends to negative infinity as \( x \) approaches 0 from the positive side.
- The exponent \( \frac{2}{\ln x} \) thus tends to 0 as \( x \) approaches 0.
### Detailed Explanation:
1. **Trigonometric Function \( \sin x \)**:
- As \( x \) approaches 0, \( \sin x \approx x \). Hence, we can approximate \( \sin x \) by \( x \).
2. **Natural Logarithm \( \ln x \)**:
- As \( x \) approaches 0 from the positive side, \( \ln x \) becomes a large negative number. Therefore, \( \frac{2}{\ln x} \) approaches 0.
3. **Expression Simplification**:
- Considering the approximation, the inner function \( (\sin x) \) can be regarded close to \( x \), and since the exponent \( \frac{2}{\ln](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff24651e9-0a43-4fcf-8170-c5e34dc6dcab%2Fe171e705-f1d1-4ff8-a6c6-ec6ecdcf83bd%2F898mwjst_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Limits Involving Trigonometric and Logarithmic Functions
In this section, we explore the behavior of a specific mathematical expression as the variable approaches a particular value.
Consider the following limit:
\[ \lim_{{x \to 0^{+}}} (\sin x)^{\frac{2}{\ln x}} \]
#### Components of the Limit Expression:
1. **\( \sin x \)**: This denotes the sine function, which is a standard trigonometric function.
2. **\( \ln x \)**: This denotes the natural logarithm of \( x \).
3. **\( x \to 0^{+} \)**: This indicates that \( x \) approaches 0 from the positive side (i.e., \( x \) remains positive).
4. **Exponent \( \frac{2}{\ln x} \)**: This part of the expression involves a division of 2 by the natural logarithm of \( x \), which becomes particularly interesting as \( x \) approaches 0 from the positive side.
#### Explanation of the Limit:
The expression analyses the behavior of \( (\sin x)^{\frac{2}{\ln x}} \) as \( x \) gets closer to 0 from the right-hand side.
- The sine function, \( \sin x \), oscillates between -1 and 1.
- The natural logarithm \( \ln x \) tends to negative infinity as \( x \) approaches 0 from the positive side.
- The exponent \( \frac{2}{\ln x} \) thus tends to 0 as \( x \) approaches 0.
### Detailed Explanation:
1. **Trigonometric Function \( \sin x \)**:
- As \( x \) approaches 0, \( \sin x \approx x \). Hence, we can approximate \( \sin x \) by \( x \).
2. **Natural Logarithm \( \ln x \)**:
- As \( x \) approaches 0 from the positive side, \( \ln x \) becomes a large negative number. Therefore, \( \frac{2}{\ln x} \) approaches 0.
3. **Expression Simplification**:
- Considering the approximation, the inner function \( (\sin x) \) can be regarded close to \( x \), and since the exponent \( \frac{2}{\ln
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