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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Question
Determine the limit using L'Hopital's Rule
![### Limit Calculation for Rational Function
This is a mathematical expression representing the limit of a rational function as \( x \) approaches 1.
\[ \lim_{{x \to 1}} \frac{\ln(x) - x + 1}{x^2 - 2x + 1} \]
**Explanation:**
- The limit notation, \(\lim_{{x \to 1}}\), indicates that we are interested in finding the value that the function approaches as \( x \) gets arbitrarily close to 1.
- The numerator of the function is \(\ln(x) - x + 1\):
- \(\ln(x)\) is the natural logarithm of \( x \).
- \( -x \) indicates that \( x \) is subtracted from \(\ln(x)\).
- The addition of 1 completes the expression.
- The denominator is \(x^2 - 2x + 1\):
- \( x^2 \) represents \( x \) squared.
- \( -2x \) indicates that twice \( x \) is subtracted from \( x^2 \).
- The addition of 1 completes the expression, forming a perfect square trinomial \((x - 1)^2\).
This expression can be useful for exploring concepts in calculus such as continuous functions, limits, and L'Hopital's Rule. When the form of the limit leads to an indeterminate form like \(\frac{0}{0}\), additional techniques, such as factoring, L'Hopital's Rule, or Taylor series expansion, might be employed to evaluate the limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ba480c0-c71f-4c99-9940-8bb79a609986%2F31c5abf6-4d82-495d-b660-7873c0818c38%2Fqd07xd.png&w=3840&q=75)
Transcribed Image Text:### Limit Calculation for Rational Function
This is a mathematical expression representing the limit of a rational function as \( x \) approaches 1.
\[ \lim_{{x \to 1}} \frac{\ln(x) - x + 1}{x^2 - 2x + 1} \]
**Explanation:**
- The limit notation, \(\lim_{{x \to 1}}\), indicates that we are interested in finding the value that the function approaches as \( x \) gets arbitrarily close to 1.
- The numerator of the function is \(\ln(x) - x + 1\):
- \(\ln(x)\) is the natural logarithm of \( x \).
- \( -x \) indicates that \( x \) is subtracted from \(\ln(x)\).
- The addition of 1 completes the expression.
- The denominator is \(x^2 - 2x + 1\):
- \( x^2 \) represents \( x \) squared.
- \( -2x \) indicates that twice \( x \) is subtracted from \( x^2 \).
- The addition of 1 completes the expression, forming a perfect square trinomial \((x - 1)^2\).
This expression can be useful for exploring concepts in calculus such as continuous functions, limits, and L'Hopital's Rule. When the form of the limit leads to an indeterminate form like \(\frac{0}{0}\), additional techniques, such as factoring, L'Hopital's Rule, or Taylor series expansion, might be employed to evaluate the limit.
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