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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![### Calculus: Finding Limits
Below are two problems that involve finding the limits of functions as the variable approaches infinity. Follow the steps to solve these limits.
#### Problem 1: Find the limits. Show your steps.
**(a)** \(\lim_{{x \to \infty}} \frac{{2x^3 + x + 1}}{{6x^3 + 1}}\)
**(b)** \(\lim_{{t \to \infty}} \frac{{2^t}}{{3^t + 1}}\)
For each problem, you'll need to analyze the behavior of the function as the variable \( x \) or \( t \) grows without bound.
1. **Identifying Dominant Terms:**
- In part (a), compare the terms in the numerator and denominator to identify which terms dominate as \( x \) becomes very large.
- In part (b), do the same comparison for the exponential terms as \( t \) becomes very large.
2. **Simplification:**
- Simplifying the functions by dividing by the highest power of \( x \) (for part (a)) or the base of the exponential term (for part (b)) helps to find the limit.
3. **Evaluation:**
- Once simplified, evaluate the functions while taking into account basic limit laws and properties.
Let's go through each step in detail in the following sections on this website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6484a2ba-7b6c-4144-bd9b-1f2d784a131b%2Fc502f991-dcd4-46c9-b680-c79199eb4dc0%2Fl6ammif_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus: Finding Limits
Below are two problems that involve finding the limits of functions as the variable approaches infinity. Follow the steps to solve these limits.
#### Problem 1: Find the limits. Show your steps.
**(a)** \(\lim_{{x \to \infty}} \frac{{2x^3 + x + 1}}{{6x^3 + 1}}\)
**(b)** \(\lim_{{t \to \infty}} \frac{{2^t}}{{3^t + 1}}\)
For each problem, you'll need to analyze the behavior of the function as the variable \( x \) or \( t \) grows without bound.
1. **Identifying Dominant Terms:**
- In part (a), compare the terms in the numerator and denominator to identify which terms dominate as \( x \) becomes very large.
- In part (b), do the same comparison for the exponential terms as \( t \) becomes very large.
2. **Simplification:**
- Simplifying the functions by dividing by the highest power of \( x \) (for part (a)) or the base of the exponential term (for part (b)) helps to find the limit.
3. **Evaluation:**
- Once simplified, evaluate the functions while taking into account basic limit laws and properties.
Let's go through each step in detail in the following sections on this website.
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