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lim i If X -C lim 6 f(x)g(x) exists then both lim ¿ lim & g(x) exist, regardless of X -C X -C the conditions of individual functions.

lim i If X -C lim 6 f(x)g(x) exists then both lim ¿ lim & g(x) exist, regardless of X -C X -C the conditions of individual functions.

Calculus: Early Transcendentals
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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Please say whether the statement is true or false.

**Statement j:** If \(\lim_{x \to c} f(x)g(x)\) exists, then both \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) exist, regardless of the conditions of individual functions. **Explanation:** This statement refers to a property of limits in calculus. It presents a situation where the limit of a product of two functions exists at a point as \(x\) approaches \(c\). For the product limit to exist, it implies certain conditions on the individual functions \(f(x)\) and \(g(x)\). This statement appears to be false because, typically, the existence of the limit of the product of two functions depends on the limits of each function separately. If \(\lim_{x \to c} f(x)g(x)\) exists without any further information, it does not guarantee the individual limits \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) exist. **Diagrams/Graphs:** No diagrams or graphs are associated with this statement. The statement is theoretical and relates to the conceptual understanding of limits in mathematical analysis.
Transcribed Image Text:**Statement j:** If \(\lim_{x \to c} f(x)g(x)\) exists, then both \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) exist, regardless of the conditions of individual functions. **Explanation:** This statement refers to a property of limits in calculus. It presents a situation where the limit of a product of two functions exists at a point as \(x\) approaches \(c\). For the product limit to exist, it implies certain conditions on the individual functions \(f(x)\) and \(g(x)\). This statement appears to be false because, typically, the existence of the limit of the product of two functions depends on the limits of each function separately. If \(\lim_{x \to c} f(x)g(x)\) exists without any further information, it does not guarantee the individual limits \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) exist. **Diagrams/Graphs:** No diagrams or graphs are associated with this statement. The statement is theoretical and relates to the conceptual understanding of limits in mathematical analysis.
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