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...
Related questions
Concept explainers
Power Operation
Power operation is topic of algebra in Math. It is use to represent repeated multiplication. Very big number and very small number can be easily express using power operation. Power operation is useful in many fields. In space engineering, it helps in representing the distance or size of particular heavenly body. In medical field, it is used to represent very small size. In medical field it helps to mention size of bacteria or virus.
Exponents
The exponent or power or index of a variable/number is the number of times that variable/number is multiplied by itself.
Question
![### Limit Calculation of Trigonometric Functions
Consider the following limit problem:
\[34) \lim_{x \to 0} \frac{\sin 3x \cot 4x}{\cot 5x}\]
We aim to evaluate this limit as \( x \) approaches 0.
In this expression:
- \(\sin 3x\) represents the sine function with argument \(3x\).
- \(\cot 4x\) represents the cotangent function with argument \(4x\), which is equivalent to \(\frac{\cos 4x}{\sin 4x}\).
- \(\cot 5x\) represents the cotangent function with argument \(5x\), which is equivalent to \(\frac{\cos 5x}{\sin 5x}\).
To solve this limit, a common technique is to simplify the expression using trigonometric identities and limit properties. For instance, \(\cot\) can be converted to \(\cos / \sin\), and \(\sin\) functions can be handled with standard small-angle approximations like \(\sin y \approx y\) when \(y\) is near 0.
The goal is to break down and simplify the expression to a manageable form that can be directly evaluated as \(x\) approaches 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e53b58-3135-4885-921f-83577fff455d%2F2d03f0dc-f0db-4ba1-a8e9-1f5d2bc79603%2F5pyknco.jpeg&w=3840&q=75)
Transcribed Image Text:
### Limit Calculation of Trigonometric Functions
Consider the following limit problem:
\[34) \lim_{x \to 0} \frac{\sin 3x \cot 4x}{\cot 5x}\]
We aim to evaluate this limit as \( x \) approaches 0.
In this expression:
- \(\sin 3x\) represents the sine function with argument \(3x\).
- \(\cot 4x\) represents the cotangent function with argument \(4x\), which is equivalent to \(\frac{\cos 4x}{\sin 4x}\).
- \(\cot 5x\) represents the cotangent function with argument \(5x\), which is equivalent to \(\frac{\cos 5x}{\sin 5x}\).
To solve this limit, a common technique is to simplify the expression using trigonometric identities and limit properties. For instance, \(\cot\) can be converted to \(\cos / \sin\), and \(\sin\) functions can be handled with standard small-angle approximations like \(\sin y \approx y\) when \(y\) is near 0.
The goal is to break down and simplify the expression to a manageable form that can be directly evaluated as \(x\) approaches 0.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning