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1. Find the limits. Show your steps. (a) lim x48 2x³ + x + 1 6x³+1 2t (b) lim tx 3 + 1

1. Find the limits. Show your steps. (a) lim x48 2x³ + x + 1 6x³+1 2t (b) lim tx 3 + 1

Calculus: Early Transcendentals
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.
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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