Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.R.73
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ ln ((x +1) / (x-1))
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Identify the form of the limit as x approaches infinity: lim_(x→∞) ln((x + 1) / (x - 1)). This is an indeterminate form of type ∞/∞.
Apply l'Hôpital's Rule, which is used to evaluate limits of indeterminate forms. First, differentiate the numerator and the denominator of the argument of the logarithm separately.
Differentiate the numerator (x + 1) to get 1, and the denominator (x - 1) to get 1. The limit now becomes lim_(x→∞) ln(1).
Since ln(1) is a constant, the limit simplifies to 0.
Conclude that the limit of the original expression as x approaches infinity is 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior at points where it may not be explicitly defined, such as at infinity or discontinuities. Evaluating limits is crucial for determining the continuity and differentiability of functions.
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Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in growth and decay problems, and is often used in limits involving exponential functions. Understanding its properties, such as its domain and range, is essential for evaluating limits that involve logarithmic expressions.
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l'Hôpital's Rule
l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits, especially when dealing with complex functions.
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