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
#### Problem Statement:
Find the number \( a \) such that the limit exists.
\[ \lim_{{x \to -2}} \frac{3x^2 + ax + a + 3}{x^2 + x - 2} \]
#### Solution Steps:
1. **Identify the numerator and denominator:**
- Numerator: \( 3x^2 + ax + a + 3 \)
- Denominator: \( x^2 + x - 2 \)
2. **Ensure the denominator is factorable:**
- Factor the denominator: \( x^2 + x - 2 = (x + 2)(x - 1) \)
3. **Find the value of \( a \) such that the limit exists:**
- The denominator equals zero at \( x = -2 \), indicating a potential discontinuity.
- Substitute \( x = -2 \) into the numerator and solve it to cancel out with the factor in the denominator. Simplify to find \( a \).
#### Input:
- Enter the value for \( a \) that makes the function continuous (i.e., the numerator and denominator cancel out the same factor).
\[ a = \, \]
#### Find the Value of the Limit:
1. **Evaluate the numerator and denominator after substituting \( x = -2 \) with the found value of \( a \).**
2. **Simplify the resulting fraction to find the limit value.**
\[ \text{Limit Value} = \, \]
### Resources:
- If you need additional help, use the provided resources: "Read It" or "Watch It".
**Need Help?**
- [Read It]
- [Watch It]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb6a032c6-0579-4de1-aa0d-b6dc8cdc2ee4%2Fa7055fa4-491c-4316-9dc0-89421b353354%2Fel6cw4_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus: Finding Limits
#### Problem Statement:
Find the number \( a \) such that the limit exists.
\[ \lim_{{x \to -2}} \frac{3x^2 + ax + a + 3}{x^2 + x - 2} \]
#### Solution Steps:
1. **Identify the numerator and denominator:**
- Numerator: \( 3x^2 + ax + a + 3 \)
- Denominator: \( x^2 + x - 2 \)
2. **Ensure the denominator is factorable:**
- Factor the denominator: \( x^2 + x - 2 = (x + 2)(x - 1) \)
3. **Find the value of \( a \) such that the limit exists:**
- The denominator equals zero at \( x = -2 \), indicating a potential discontinuity.
- Substitute \( x = -2 \) into the numerator and solve it to cancel out with the factor in the denominator. Simplify to find \( a \).
#### Input:
- Enter the value for \( a \) that makes the function continuous (i.e., the numerator and denominator cancel out the same factor).
\[ a = \, \]
#### Find the Value of the Limit:
1. **Evaluate the numerator and denominator after substituting \( x = -2 \) with the found value of \( a \).**
2. **Simplify the resulting fraction to find the limit value.**
\[ \text{Limit Value} = \, \]
### Resources:
- If you need additional help, use the provided resources: "Read It" or "Watch It".
**Need Help?**
- [Read It]
- [Watch It]
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