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.

Let us denote f(x)=5 if x<215 if x≥2g(x)=15 if x<25 if x≥2Then limx→2f(x)=does not exist and limx→2g(x)=does not exist But, f(x)g(x)=5×15 if x<215×5 if x≥2=1 if x<21 if x≥2=1 for all xf(x)g(x)=1This means limx→2f(x)g(x)=1 exists. But limx→2f(x) and limx→2g(x) did not exist. So the given statement is false. Answer: False

Answered: lim i If X -C lim 6 f(x)g(x) exists…

Math

Calculus

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

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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