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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![The image contains two mathematical limit expressions related to natural logarithms and trigonometric functions. Below is the transcription and a detailed explanation suitable for an educational website:
---
**Mathematical Limits involving Natural Logarithms and Trigonometric Functions**
In this section, we will explore the behavior of natural logarithmic functions involving trigonometric expressions as the variable \( x \) approaches a specific value from the left. We consider the following limits:
1. **Limit of the Natural Logarithm of the Tangent Function**
\[
\lim_{x \to \pi^-} \ln(\tan x)
\]
This expression is concerned with the limit of the natural logarithm of the tangent function, \(\ln(\tan x)\), as the variable \( x \) approaches \(\pi\) from the left side (\(\pi^-\)).
2. **Limit of the Natural Logarithm of the Cosecant Function**
\[
\lim_{x \to \pi^-} \ln(\csc x)
\]
This expression focuses on the limit of the natural logarithm of the cosecant function, \(\ln(\csc x)\), as the variable \( x \) approaches \(\pi\) from the left side (\(\pi^-\)).
**Conceptual Explanation:**
- **Tangent Function (\(\tan x\)):** Tangent is periodic with singularities (undefined points) at \(x = \frac{\pi}{2} + k\pi\) for \(k \in \mathbb{Z}\). As \( x \) approaches \(\pi\) from the left, \(\tan x\) approaches zero.
- **Cosecant Function (\(\csc x\)):** The cosecant function, which is the reciprocal of the sine function (\(\csc x = \frac{1}{\sin x}\)), has singularities at integer multiples of \(\pi\). As \( x \) approaches \(\pi\) from the left, \(\sin x\) approaches zero, making \(\csc x\) approach infinity.
These detailed considerations assist in understanding how logarithmic functions behave in conjunction with trigonometric functions close to their critical points.
---
This content should help students grasp the concept of limits involving natural logarithms and trigonometric functions as they approach particular values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe7817cb-5274-4434-90ec-13ab480c973d%2Fd6b5e2cf-b145-467f-81fd-cb04b89dcafc%2Fmdhkbsh_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains two mathematical limit expressions related to natural logarithms and trigonometric functions. Below is the transcription and a detailed explanation suitable for an educational website:
---
**Mathematical Limits involving Natural Logarithms and Trigonometric Functions**
In this section, we will explore the behavior of natural logarithmic functions involving trigonometric expressions as the variable \( x \) approaches a specific value from the left. We consider the following limits:
1. **Limit of the Natural Logarithm of the Tangent Function**
\[
\lim_{x \to \pi^-} \ln(\tan x)
\]
This expression is concerned with the limit of the natural logarithm of the tangent function, \(\ln(\tan x)\), as the variable \( x \) approaches \(\pi\) from the left side (\(\pi^-\)).
2. **Limit of the Natural Logarithm of the Cosecant Function**
\[
\lim_{x \to \pi^-} \ln(\csc x)
\]
This expression focuses on the limit of the natural logarithm of the cosecant function, \(\ln(\csc x)\), as the variable \( x \) approaches \(\pi\) from the left side (\(\pi^-\)).
**Conceptual Explanation:**
- **Tangent Function (\(\tan x\)):** Tangent is periodic with singularities (undefined points) at \(x = \frac{\pi}{2} + k\pi\) for \(k \in \mathbb{Z}\). As \( x \) approaches \(\pi\) from the left, \(\tan x\) approaches zero.
- **Cosecant Function (\(\csc x\)):** The cosecant function, which is the reciprocal of the sine function (\(\csc x = \frac{1}{\sin x}\)), has singularities at integer multiples of \(\pi\). As \( x \) approaches \(\pi\) from the left, \(\sin x\) approaches zero, making \(\csc x\) approach infinity.
These detailed considerations assist in understanding how logarithmic functions behave in conjunction with trigonometric functions close to their critical points.
---
This content should help students grasp the concept of limits involving natural logarithms and trigonometric functions as they approach particular values.
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