lim(-x² + x-2) x-2
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![### Limits of Rational Functions
Below are several examples of limits involving rational functions:
1. **Limit as \( x \) Approaches 2:**
\[
\lim_{{x \to 2}} (-x^2 + x - 2)
\]
We evaluate the limit of the quadratic function \( -x^2 + x - 2 \) as \( x \) approaches 2.
2. **Limit as \( x \) Approaches -4:**
\[
\lim_{{x \to -4}} \frac{{x^2 + 5x + 4}}{{x + 2}}
\]
This is the limit of a rational function where the numerator is \( x^2 + 5x + 4 \) and the denominator is \( x + 2 \), as \( x \) approaches -4.
3. **Limit as \( x \) Approaches 0:**
\[
\lim_{{x \to 0}} \frac{{x^2 + 7x + 6}}{{x + 3}}
\]
Evaluating the limit of the rational function \( \frac{{x^2 + 7x + 6}}{{x + 3}} \) as \( x \) approaches 0.
4. **Limit as \( x \) Approaches -3:**
\[
\lim_{{x \to -3}} \frac{{x^2 + x - 6}}{{x^2 + 8x + 15}}
\]
Finding the limit of the rational function \( \frac{{x^2 + x - 6}}{{x^2 + 8x + 15}} \) as \( x \) approaches -3.
### Explanation and Techniques
When evaluating limits involving polynomial and rational functions, techniques such as direct substitution, factoring, and simplification are commonly used. If direct substitution results in an indeterminate form, further simplification such as factoring the quadratic equation might be needed to resolve the limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d31ce46-b2af-402e-a134-bb82133eeb7f%2Fabad5183-646b-41fb-a860-40671fe10439%2Fvbfkdc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Limits of Rational Functions
Below are several examples of limits involving rational functions:
1. **Limit as \( x \) Approaches 2:**
\[
\lim_{{x \to 2}} (-x^2 + x - 2)
\]
We evaluate the limit of the quadratic function \( -x^2 + x - 2 \) as \( x \) approaches 2.
2. **Limit as \( x \) Approaches -4:**
\[
\lim_{{x \to -4}} \frac{{x^2 + 5x + 4}}{{x + 2}}
\]
This is the limit of a rational function where the numerator is \( x^2 + 5x + 4 \) and the denominator is \( x + 2 \), as \( x \) approaches -4.
3. **Limit as \( x \) Approaches 0:**
\[
\lim_{{x \to 0}} \frac{{x^2 + 7x + 6}}{{x + 3}}
\]
Evaluating the limit of the rational function \( \frac{{x^2 + 7x + 6}}{{x + 3}} \) as \( x \) approaches 0.
4. **Limit as \( x \) Approaches -3:**
\[
\lim_{{x \to -3}} \frac{{x^2 + x - 6}}{{x^2 + 8x + 15}}
\]
Finding the limit of the rational function \( \frac{{x^2 + x - 6}}{{x^2 + 8x + 15}} \) as \( x \) approaches -3.
### Explanation and Techniques
When evaluating limits involving polynomial and rational functions, techniques such as direct substitution, factoring, and simplification are commonly used. If direct substitution results in an indeterminate form, further simplification such as factoring the quadratic equation might be needed to resolve the limit.
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