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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![### Problem Statement:
If \( f \) and \( g \) are continuous functions with \( f(5) = 2 \) and
\[
\lim_{x \to 5} [2f(x) - g(x)] = 5
\]
find \( g(5) \).
### Multiple Choice Options:
- \( \circ \) 1
- \( \circ \) -1
- \( \circ \) 0
- \( \circ \) 5
### Solution Explanation:
To find \( g(5) \), we utilize the given information:
1. **Given Continuity**: Since \( f \) and \( g \) are continuous functions, \( \lim_{x \to 5} f(x) = f(5) = 2 \) and \( \lim_{x \to 5} g(x) = g(5) \).
2. **Substitute Known Values in the Limit**:
\[
\lim_{x \to 5} [2f(x) - g(x)] = 5
\]
Using \( \lim_{x \to 5} f(x) = 2 \), we substitute:
\[
2 \times 2 - g(5) = 5
\]
\[
4 - g(5) = 5
\]
\[
g(5) = -1
\]
Thus, the correct answer is \( \circ \) -1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdb03a2e-9fad-42e4-9ad5-1d8f931a6cde%2Fb2c007e1-b491-4554-af2a-f5446b9d6ffd%2F99pazb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
If \( f \) and \( g \) are continuous functions with \( f(5) = 2 \) and
\[
\lim_{x \to 5} [2f(x) - g(x)] = 5
\]
find \( g(5) \).
### Multiple Choice Options:
- \( \circ \) 1
- \( \circ \) -1
- \( \circ \) 0
- \( \circ \) 5
### Solution Explanation:
To find \( g(5) \), we utilize the given information:
1. **Given Continuity**: Since \( f \) and \( g \) are continuous functions, \( \lim_{x \to 5} f(x) = f(5) = 2 \) and \( \lim_{x \to 5} g(x) = g(5) \).
2. **Substitute Known Values in the Limit**:
\[
\lim_{x \to 5} [2f(x) - g(x)] = 5
\]
Using \( \lim_{x \to 5} f(x) = 2 \), we substitute:
\[
2 \times 2 - g(5) = 5
\]
\[
4 - g(5) = 5
\]
\[
g(5) = -1
\]
Thus, the correct answer is \( \circ \) -1.
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