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 presents a mathematical problem involving limits in vector calculus. It asks to find the limit or indicate if it does not exist for the given vector function as \( t \) approaches \( \pi \).
The function is represented as a vector:
\[
\lim_{{t \to \pi}} \left[ \left( \sin \frac{4}{3}t \right) \mathbf{i} + \left( \cos \frac{t}{6} \right) \mathbf{j} + \left( \tan \frac{3}{4}t \right) \mathbf{k} \right]
\]
This is a three-dimensional vector with components:
- For the \( \mathbf{i} \)-component: \(\sin\left(\frac{4}{3}t\right)\)
- For the \( \mathbf{j} \)-component: \(\cos\left(\frac{t}{6}\right)\)
- For the \( \mathbf{k} \)-component: \(\tan\left(\frac{3}{4}t\right)\)
The task is to determine each component's limit as \( t \) approaches \( \pi \) and evaluate if the overall vector limit exists.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F671be518-f9e6-4df9-9304-33791fefeccb%2F05306d2f-2739-4592-b825-5c8b3f29cb3a%2F74hu5f5_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a mathematical problem involving limits in vector calculus. It asks to find the limit or indicate if it does not exist for the given vector function as \( t \) approaches \( \pi \).
The function is represented as a vector:
\[
\lim_{{t \to \pi}} \left[ \left( \sin \frac{4}{3}t \right) \mathbf{i} + \left( \cos \frac{t}{6} \right) \mathbf{j} + \left( \tan \frac{3}{4}t \right) \mathbf{k} \right]
\]
This is a three-dimensional vector with components:
- For the \( \mathbf{i} \)-component: \(\sin\left(\frac{4}{3}t\right)\)
- For the \( \mathbf{j} \)-component: \(\cos\left(\frac{t}{6}\right)\)
- For the \( \mathbf{k} \)-component: \(\tan\left(\frac{3}{4}t\right)\)
The task is to determine each component's limit as \( t \) approaches \( \pi \) and evaluate if the overall vector limit exists.
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