Textbook Question
Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.
lim x→∞ cot^−1
Ch. 2 - Limits
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 2, Problem 82b
Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.
lim x→−∞ cot^−1x
Verified step by step guidance1
Step 1: Understand the function \( y = \cot^{-1}(x) \). The inverse cotangent function, \( \cot^{-1}(x) \), is the angle whose cotangent is \( x \). It is defined for all real numbers and its range is \((0, \pi)\).
Step 2: Consider the behavior of \( \cot^{-1}(x) \) as \( x \to -\infty \). As \( x \) becomes very large in the negative direction, the angle whose cotangent is \( x \) approaches \( \pi \).
Step 3: Visualize the graph of \( y = \cot^{-1}(x) \). As \( x \to -\infty \), the graph approaches the horizontal asymptote at \( y = \pi \).
Step 4: Use the graph to determine the limit. The graph shows that as \( x \to -\infty \), \( \cot^{-1}(x) \) approaches \( \pi \).
Step 5: Conclude that \( \lim_{x \to -\infty} \cot^{-1}(x) = \pi \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Cotangent Function
The inverse cotangent function, denoted as cot^−1(x), is the function that returns the angle whose cotangent is x. It is defined for all real numbers and has a range of (0, π). Understanding this function is crucial for analyzing its behavior as x approaches different limits.
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Limits in Calculus
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Evaluating limits helps in understanding the behavior of functions at boundaries, including infinity. In this case, we are interested in the limit of cot^−1(x) as x approaches negative infinity.
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Behavior of Functions at Infinity
The behavior of functions as they approach infinity or negative infinity is essential for understanding their long-term trends. For the cotangent function, as x approaches negative infinity, the value of cot^−1(x) approaches π. This concept helps in predicting the output of the function without direct computation.
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Related Practice
Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).
f(x)=cos x+2√x / √x.
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Textbook Question
Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
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Textbook Question
Use an appropriate limit definition to prove the following limits.
lim x→1 (5x−2) =3;
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Textbook Question
a. Use the Intermediate Value Theorem to show that the equation has a solution in the given interval.
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Textbook Question
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