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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![**Problem Statement:**
Calculate the following limit:
\[
\lim_{{x \to 0}} [\csc(x) - \cot(x)].
\]
**Explanation:**
This mathematical expression asks to find the limit of the function \(\csc(x) - \cot(x)\) as \(x\) approaches 0.
- \(\csc(x)\) is the cosecant function, defined as \(1/\sin(x)\).
- \(\cot(x)\) is the cotangent function, defined as \(\cos(x)/\sin(x)\).
**Objective:**
The goal is to determine what value this expression approaches as \(x\) gets arbitrarily close to 0, from the right side (since \(\csc(x)\) and \(\cot(x)\) are not defined at \(x = 0\)).
**Steps for Evaluation:**
1. Substitute the definitions of \(\csc(x)\) and \(\cot(x)\) into the expression.
2. Simplify the resulting expression.
3. Determine the limit using algebraic manipulation, L'Hôpital's rule, or a series expansion, ensuring that all steps mathematically justify the behavior of the function as \(x\) approaches 0.
**Note:** Students should be familiar with trigonometric identities and limit concepts to efficiently solve this problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3cf874b-7a7b-478f-a7b8-421442e72224%2F05fcb65e-643a-4150-b89f-bedf23a8d5d8%2Fcay6z4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Calculate the following limit:
\[
\lim_{{x \to 0}} [\csc(x) - \cot(x)].
\]
**Explanation:**
This mathematical expression asks to find the limit of the function \(\csc(x) - \cot(x)\) as \(x\) approaches 0.
- \(\csc(x)\) is the cosecant function, defined as \(1/\sin(x)\).
- \(\cot(x)\) is the cotangent function, defined as \(\cos(x)/\sin(x)\).
**Objective:**
The goal is to determine what value this expression approaches as \(x\) gets arbitrarily close to 0, from the right side (since \(\csc(x)\) and \(\cot(x)\) are not defined at \(x = 0\)).
**Steps for Evaluation:**
1. Substitute the definitions of \(\csc(x)\) and \(\cot(x)\) into the expression.
2. Simplify the resulting expression.
3. Determine the limit using algebraic manipulation, L'Hôpital's rule, or a series expansion, ensuring that all steps mathematically justify the behavior of the function as \(x\) approaches 0.
**Note:** Students should be familiar with trigonometric identities and limit concepts to efficiently solve this problem.
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