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2. Compute the following limits: (a) lim -x+ 7x + 3 (b) lim -x* + 7x +3 3 + 7x + 3 (c) lim

2. Compute the following limits: (a) lim -x+ 7x + 3 (b) lim -x* + 7x +3 3 + 7x + 3 (c) lim

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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## Exercise: Compute the Following Limits 1. **Problem Statement** Evaluate the limits in each of the following expressions: (a) \( \lim_{{x \to \infty}} (-x^4 + 7x^3 + 3) \) (b) \( \lim_{{x \to -\infty}} (-x^4 + 7x + 3) \) (c) \( \lim_{{x \to -\infty}} \frac{x^3 + 7x + 3}{x} \) (d) \( \lim_{{x \to \infty}} \frac{x^2 + 3}{x^3} \) 2. **Solution Discussion** Each of these problems involves taking the limit of a polynomial or rational function as \(x\) approaches infinity or negative infinity. To solve these limit problems, you should analyze the degrees of the polynomials and determine the dominant term. Look out for cases where the highest-degree term will dictate the behavior of the limit. - For part (a) and (b), notice that the term \( -x^4 \) has the highest degree. Consider how this term behaves as \( x \) approaches either positive or negative infinity. - For part (c), divide each term in the numerator and the denominator by \(x\) and then determine the behavior of the expression. - For part (d), consider the relative degrees of the numerator and denominator to determine how the fraction behaves as \( x \) goes to infinity. 3. **Graphical Visualization** (if applicable) Since this is a problem purely involving algebraic manipulation and understanding of limits, a graph may not be provided, but sketching the functions may help in visualizing how they behave as \( x \) approaches infinity or negative infinity.
Transcribed Image Text:## Exercise: Compute the Following Limits 1. **Problem Statement** Evaluate the limits in each of the following expressions: (a) \( \lim_{{x \to \infty}} (-x^4 + 7x^3 + 3) \) (b) \( \lim_{{x \to -\infty}} (-x^4 + 7x + 3) \) (c) \( \lim_{{x \to -\infty}} \frac{x^3 + 7x + 3}{x} \) (d) \( \lim_{{x \to \infty}} \frac{x^2 + 3}{x^3} \) 2. **Solution Discussion** Each of these problems involves taking the limit of a polynomial or rational function as \(x\) approaches infinity or negative infinity. To solve these limit problems, you should analyze the degrees of the polynomials and determine the dominant term. Look out for cases where the highest-degree term will dictate the behavior of the limit. - For part (a) and (b), notice that the term \( -x^4 \) has the highest degree. Consider how this term behaves as \( x \) approaches either positive or negative infinity. - For part (c), divide each term in the numerator and the denominator by \(x\) and then determine the behavior of the expression. - For part (d), consider the relative degrees of the numerator and denominator to determine how the fraction behaves as \( x \) goes to infinity. 3. **Graphical Visualization** (if applicable) Since this is a problem purely involving algebraic manipulation and understanding of limits, a graph may not be provided, but sketching the functions may help in visualizing how they behave as \( x \) approaches infinity or negative infinity.
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