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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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 82a
Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.
lim x→∞ cot^−1
Verified step by step guidance1
Identify the function: The function given is \( y = \cot^{-1}(x) \), which is the inverse cotangent function.
Understand the behavior of \( \cot^{-1}(x) \): As \( x \to \infty \), the inverse cotangent function approaches a specific value.
Recall the range of \( \cot^{-1}(x) \): The range of \( \cot^{-1}(x) \) is \((0, \pi)\).
Analyze the limit: As \( x \to \infty \), the value of \( \cot^{-1}(x) \) approaches the lower bound of its range.
Conclude the limit: Based on the behavior of the inverse cotangent function, determine the limit as \( x \to \infty \).
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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 and limits, especially as x approaches infinity.
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Derivatives of Other Inverse Trigonometric Functions
Limits in Calculus
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. In this context, evaluating the limit of cot^−1(x) as x approaches infinity helps determine the horizontal asymptote of the function, which is essential for understanding its long-term behavior.
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Graphical Interpretation of Limits
Graphical interpretation of limits involves analyzing the graph of a function to determine its behavior as the input approaches a specific value. For cot^−1(x), examining the graph as x approaches infinity allows us to visually assess the limit and understand how the function behaves at extreme values.
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6:47Finding Limits Numerically and Graphically
Related Practice
Textbook Question
Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x)=cos x+2√x / √x.
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)=3e^x+10 / e^x
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Textbook Question
a. Use the Intermediate Value Theorem to show that the equation has a solution in the given interval.
x=cos x; (0,π/2)
Textbook Question
A sine limit It can be shown that 1−x^2/ 6 ≤ sin x/ x ≤1, for x near 0.
Use these inequalities to evaluate lim x→0 sin x/ x.
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^−1x
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