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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![**Mathematics Problem Verification**
---
**Question:**
(a) If \( f \) is continuous and \( f(5) = 2 \) and \( f(4) = 3 \), then \[\lim_{{x \to 2}} f(4x^2 - 11) = 2.\]
**Options:**
- True
- False
---
In this problem, we are given that a function \( f \) is continuous, and we know two of its values: \( f(5) = 2 \) and \( f(4) = 3 \). The task is to determine whether the following limit statement is true: \[\lim_{{x \to 2}} f(4x^2 - 11) = 2.\]
To solve this, we need to check whether the input of the function \( f \) approaches the value 5 as \( x \) approaches 2, given the expression inside the limit.
---
### Solution Steps:
1. **Evaluate the Expression Inside \( f \):**
\[4x^2 - 11\]
2. **Substitute \( x = 2 \):**
\[4(2)^2 - 11 = 4(4) - 11 = 16 - 11 = 5\]
Thus, as \( x \) approaches 2, the expression \( 4x^2 - 11 \) approaches 5.
3. **Limit of the Composition**:
Since \( f \) is continuous, we can directly substitute the limiting value inside \( f \):
\[\lim_{{x \to 2}} f(4x^2 - 11) = f(\lim_{{x \to 2}} (4x^2 - 11)) = f(5)\]
4. **Given Value**:
\[f(5) = 2\]
Hence,
\[\lim_{{x \to 2}} f(4x^2 - 11) = 2\]
Thus, the statement is **True**.
---
**Answer:**
- True](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc931b694-aaff-47c2-895e-73bc1d52993a%2F27eaa1d0-9593-4e1c-a01c-b3b884c834e2%2Fpb926t_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematics Problem Verification**
---
**Question:**
(a) If \( f \) is continuous and \( f(5) = 2 \) and \( f(4) = 3 \), then \[\lim_{{x \to 2}} f(4x^2 - 11) = 2.\]
**Options:**
- True
- False
---
In this problem, we are given that a function \( f \) is continuous, and we know two of its values: \( f(5) = 2 \) and \( f(4) = 3 \). The task is to determine whether the following limit statement is true: \[\lim_{{x \to 2}} f(4x^2 - 11) = 2.\]
To solve this, we need to check whether the input of the function \( f \) approaches the value 5 as \( x \) approaches 2, given the expression inside the limit.
---
### Solution Steps:
1. **Evaluate the Expression Inside \( f \):**
\[4x^2 - 11\]
2. **Substitute \( x = 2 \):**
\[4(2)^2 - 11 = 4(4) - 11 = 16 - 11 = 5\]
Thus, as \( x \) approaches 2, the expression \( 4x^2 - 11 \) approaches 5.
3. **Limit of the Composition**:
Since \( f \) is continuous, we can directly substitute the limiting value inside \( f \):
\[\lim_{{x \to 2}} f(4x^2 - 11) = f(\lim_{{x \to 2}} (4x^2 - 11)) = f(5)\]
4. **Given Value**:
\[f(5) = 2\]
Hence,
\[\lim_{{x \to 2}} f(4x^2 - 11) = 2\]
Thus, the statement is **True**.
---
**Answer:**
- True
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