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:**
Find the indicated limit given that
\[ \lim_{{x \to a}} f(x) = 5 \quad \text{and} \quad \lim_{{x \to a}} g(x) = 4. \]
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] \]
___
**Detailed Explanation:**
To solve this limit problem, we will use the properties of limits. Specifically, we can use the Linear Combination of Limits theorem which states that if \(\lim_{{x \to a}} f(x) = L\) and \(\lim_{{x \to a}} g(x) = M\), then for any constants \(c_1\) and \(c_2\):
\[ \lim_{{x \to a}} [c_1 f(x) + c_2 g(x)] = c_1 \lim_{{x \to a}} f(x) + c_2 \lim_{{x \to a}} g(x) \]
Given:
\[ \lim_{{x \to a}} f(x) = 5 \]
\[ \lim_{{x \to a}} g(x) = 4 \]
We need to find:
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] \]
By applying the Linear Combination of Limits theorem:
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] = 2 \lim_{{x \to a}} f(x) - 4 \lim_{{x \to a}} g(x) \]
Substitute the given limits into the equation:
\[ = 2 \cdot 5 - 4 \cdot 4 \]
\[ = 10 - 16 \]
\[ = -6 \]
So,
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] = -6 \]
___
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Transcribed Image Text:**Problem Statement:**
Find the indicated limit given that
\[ \lim_{{x \to a}} f(x) = 5 \quad \text{and} \quad \lim_{{x \to a}} g(x) = 4. \]
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] \]
___
**Detailed Explanation:**
To solve this limit problem, we will use the properties of limits. Specifically, we can use the Linear Combination of Limits theorem which states that if \(\lim_{{x \to a}} f(x) = L\) and \(\lim_{{x \to a}} g(x) = M\), then for any constants \(c_1\) and \(c_2\):
\[ \lim_{{x \to a}} [c_1 f(x) + c_2 g(x)] = c_1 \lim_{{x \to a}} f(x) + c_2 \lim_{{x \to a}} g(x) \]
Given:
\[ \lim_{{x \to a}} f(x) = 5 \]
\[ \lim_{{x \to a}} g(x) = 4 \]
We need to find:
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] \]
By applying the Linear Combination of Limits theorem:
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] = 2 \lim_{{x \to a}} f(x) - 4 \lim_{{x \to a}} g(x) \]
Substitute the given limits into the equation:
\[ = 2 \cdot 5 - 4 \cdot 4 \]
\[ = 10 - 16 \]
\[ = -6 \]
So,
\[ \lim_{{x \to a}} [2f(x) - 4g(x)] = -6 \]
___
**Need Help?**
- **Read It:** Provides additional reading resources and examples.
- **Master It:** Offers a step-by-step guide to mastering this type of problem.
- **Talk to a Tutor:** Connects you with a tutor for one-on-one assistance.
These resources can help you further understand the concept and solution process for this limit problem.
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