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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![**Finding the Limit Using L'Hôpital's Rule**
**Problem Statement:**
Find the indicated limits, noting L'Hôpital's Rule wherever it is used.
\[ \lim_ {x \to \infty} \left( 1 - \frac{a}{x} \right)^x \text{ (constant \( a \))} \]
**Explanation:**
This limit problem requires applying advanced calculus techniques. The expression inside the limit can be simplified with the use of exponential functions and natural logarithms. In particular, recognizing the indeterminate form \((1 + \text{small number})^{\text{large number}}\), we can use the exponential function with Euler's number \( e \), and sometimes L'Hôpital's Rule is involved when encountering indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
1. **Transform the Exponential Form:**
Start by taking the natural logarithm of the given function to handle the exponent:
\[ y = \left( 1 - \frac{a}{x} \right)^x \]
\[ \ln y = \ln \left( \left( 1 - \frac{a}{x} \right)^x \right) \]
\[ \ln y = x \ln \left( 1 - \frac{a}{x} \right) \]
2. **Evaluate the Limit Inside the Logarithm:**
Now consider the limit of the exponent expression:
\[ \lim_{x \to \infty} x \ln \left( 1 - \frac{a}{x} \right) \]
To evaluate this limit, notice the form \( x \ln \left( 1 - \frac{a}{x} \right) \) becomes \( \infty \cdot (-0) \), an indeterminate form that can be approached using tools such as series expansion or L'Hôpital's Rule.
3. **Using a Series Expansion:**
For large \( x \):
\[ \ln \left( 1 - \frac{a}{x} \right) \approx -\frac{a}{x} \text{ (using the first term of the Taylor series expansion around 0)} \]
Thus:
\[ x \ln \left( 1 - \frac{a}{x} \right) \approx x \left( -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ad87420-52d2-4e34-815e-eb9bcb5c95c0%2F06ecef5e-18f1-49f6-a659-993f22f89110%2F8uwlco_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Limit Using L'Hôpital's Rule**
**Problem Statement:**
Find the indicated limits, noting L'Hôpital's Rule wherever it is used.
\[ \lim_ {x \to \infty} \left( 1 - \frac{a}{x} \right)^x \text{ (constant \( a \))} \]
**Explanation:**
This limit problem requires applying advanced calculus techniques. The expression inside the limit can be simplified with the use of exponential functions and natural logarithms. In particular, recognizing the indeterminate form \((1 + \text{small number})^{\text{large number}}\), we can use the exponential function with Euler's number \( e \), and sometimes L'Hôpital's Rule is involved when encountering indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
1. **Transform the Exponential Form:**
Start by taking the natural logarithm of the given function to handle the exponent:
\[ y = \left( 1 - \frac{a}{x} \right)^x \]
\[ \ln y = \ln \left( \left( 1 - \frac{a}{x} \right)^x \right) \]
\[ \ln y = x \ln \left( 1 - \frac{a}{x} \right) \]
2. **Evaluate the Limit Inside the Logarithm:**
Now consider the limit of the exponent expression:
\[ \lim_{x \to \infty} x \ln \left( 1 - \frac{a}{x} \right) \]
To evaluate this limit, notice the form \( x \ln \left( 1 - \frac{a}{x} \right) \) becomes \( \infty \cdot (-0) \), an indeterminate form that can be approached using tools such as series expansion or L'Hôpital's Rule.
3. **Using a Series Expansion:**
For large \( x \):
\[ \ln \left( 1 - \frac{a}{x} \right) \approx -\frac{a}{x} \text{ (using the first term of the Taylor series expansion around 0)} \]
Thus:
\[ x \ln \left( 1 - \frac{a}{x} \right) \approx x \left( -
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