sin 3x cot 4x lim cot 5x

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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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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### 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.
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.
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