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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Question
![### Part c.
**True or False:**
\[
\lim_{{x \to -\infty}} \frac{3e^{2x} + 5e^{-x}}{e^{2x} - e^{-x}} = 5.
\]
To explore whether this statement is true or false, we must evaluate the limit of the given expression as \( x \) approaches negative infinity.
In this expression:
- The numerator is \( 3e^{2x} + 5e^{-x} \)
- The denominator is \( e^{2x} - e^{-x} \)
As \( x \) approaches negative infinity, the exponential terms \( e^{2x} \) and \( e^{-x} \) behave differently. Specifically, \( e^{2x} \) rapidly approaches zero, and \( e^{-x} \) grows exponentially. This analysis helps to simplify and accurately evaluate the limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9a5f5a5-4eae-476c-bcab-4a001ac27552%2Fbea90819-7fba-4910-bfcc-9571c8f83a72%2Fen1nzp7_processed.png&w=3840&q=75)
Transcribed Image Text:### Part c.
**True or False:**
\[
\lim_{{x \to -\infty}} \frac{3e^{2x} + 5e^{-x}}{e^{2x} - e^{-x}} = 5.
\]
To explore whether this statement is true or false, we must evaluate the limit of the given expression as \( x \) approaches negative infinity.
In this expression:
- The numerator is \( 3e^{2x} + 5e^{-x} \)
- The denominator is \( e^{2x} - e^{-x} \)
As \( x \) approaches negative infinity, the exponential terms \( e^{2x} \) and \( e^{-x} \) behave differently. Specifically, \( e^{2x} \) rapidly approaches zero, and \( e^{-x} \) grows exponentially. This analysis helps to simplify and accurately evaluate the limit.
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