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т + 2г + 1 а) lim -1 r5 - 2r - 1

т + 2г + 1 а) lim -1 r5 - 2r - 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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Evaluate the limits below using any appropriate method. If used L'Hopital's Rule, state that you did and why it applies.

### Problem Statement Evaluate the following limit: \[ \lim_{x \to -1} \frac{x^4 + 2x + 1}{x^5 - 2x - 1} \] ### Explanation: \(x\) is approaching \(-1\) in the given limit expression. The numerator is a polynomial of degree 4, and the denominator is a polynomial of degree 5. To solve this limit, one can apply various techniques such as factoring, polynomial division, or L'Hôpital's rule if the limit presents an indeterminate form. For instance, one method is to check if factoring the polynomials simplifies the expression, or directly evaluate the function at \( x = -1 \) if it's straightforward. Feel free to continue the analysis with your preferred method or check step-by-step solutions for better understanding.
Transcribed Image Text:### Problem Statement Evaluate the following limit: \[ \lim_{x \to -1} \frac{x^4 + 2x + 1}{x^5 - 2x - 1} \] ### Explanation: \(x\) is approaching \(-1\) in the given limit expression. The numerator is a polynomial of degree 4, and the denominator is a polynomial of degree 5. To solve this limit, one can apply various techniques such as factoring, polynomial division, or L'Hôpital's rule if the limit presents an indeterminate form. For instance, one method is to check if factoring the polynomials simplifies the expression, or directly evaluate the function at \( x = -1 \) if it's straightforward. Feel free to continue the analysis with your preferred method or check step-by-step solutions for better understanding.
The provided image contains a mathematical limit expression, commonly found in calculus. **Problem (b): Evaluate the limit below** \[ \lim_{{x \to 2}} \frac{{4 - 2x}}{{x^2 - x - 6}} \] To solve this limit, consider the following steps: 1. **Factor the denominator:** - Factor the quadratic expression in the denominator, \(x^2 - x - 6\), into two binomial expressions. 2. **Simplification:** - If possible, simplify the expression by canceling out common factors in the numerator and the denominator. This limit can be used to illustrate concepts related to polynomial limits, factoring techniques, and limits involving rational functions.
Transcribed Image Text:The provided image contains a mathematical limit expression, commonly found in calculus. **Problem (b): Evaluate the limit below** \[ \lim_{{x \to 2}} \frac{{4 - 2x}}{{x^2 - x - 6}} \] To solve this limit, consider the following steps: 1. **Factor the denominator:** - Factor the quadratic expression in the denominator, \(x^2 - x - 6\), into two binomial expressions. 2. **Simplification:** - If possible, simplify the expression by canceling out common factors in the numerator and the denominator. This limit can be used to illustrate concepts related to polynomial limits, factoring techniques, and limits involving rational functions.
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