Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Review - Concept Check - Page 165: 3

Answer

(a) Sum Law: $\lim\limits_{x \to a}[f(x)+g(x)]=\lim\limits_{x \to a}f(x)+\lim\limits_{x \to a}g(x)$ (b) Difference Law: $\lim\limits_{x \to a}[f(x)-g(x)]=\lim\limits_{x \to a}f(x)-\lim\limits_{x \to a}g(x)$ (c) Constant Multiple Law: $\lim\limits_{x \to a}[cf(x)]=c\lim\limits_{x \to a}f(x)$ (d) Product Law: $\lim\limits_{x \to a}[f(x)g(x)]=\lim\limits_{x \to a}f(x)\times\lim\limits_{x \to a}g(x)$ (e) Quotient Law: $\lim\limits_{x \to a}\frac{f(x)}{g(x)}=\frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}$ Note, only if $\lim\limits_{x \to a}g(x)\ne0$ (f) Power Law: $\lim\limits_{x \to a}[f(x)]^n=[\lim\limits_{x \to a}f(x)]^n$ Note, $\underline n$ is a positive integer. (g) Root Law: $\lim\limits_{x \to a}\sqrt[n] {f(x)}=\sqrt[n] {\lim\limits_{x \to a}f(x)}$ Note, $\underline n$ is a positive integer. If $n$ is even, we assume that $\lim\limits_{x \to a}f(x) > 0$

Work Step by Step

As stated in the chapter 2.3, we know the following: Let's suppose that $c$ is a constant and the limits $\lim\limits_{x \to a}f(x)$ and $\lim\limits_{x \to a}g(x)$ exist. (a) Sum Law: $\lim\limits_{x \to a}[f(x)+g(x)]=\lim\limits_{x \to a}f(x)+\lim\limits_{x \to a}g(x)$ The limit of sum is the sum of the limits. (b) Difference Law: $\lim\limits_{x \to a}[f(x)-g(x)]=\lim\limits_{x \to a}f(x)-\lim\limits_{x \to a}g(x)$ The limit of a difference is the difference of the limits. (c) Constant Multiple Law: $\lim\limits_{x \to a}[cf(x)]=c\lim\limits_{x \to a}f(x)$ The limit of a constant times a function is the constant times the limit of the function. (d) Product Law: $\lim\limits_{x \to a}[f(x)g(x)]=\lim\limits_{x \to a}f(x)\times\lim\limits_{x \to a}g(x)$ The limit of a product is the product of the limits. (e) Quotient Law: $\lim\limits_{x \to a}\frac{f(x)}{g(x)}=\frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}$ Note, only if $\lim\limits_{x \to a}g(x)\ne0$ The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0). (f) Power Law: $\lim\limits_{x \to a}[f(x)]^n=[\lim\limits_{x \to a}f(x)]^n$ Note, $\underline n$ is a positive integer. (g) Root Law: $\lim\limits_{x \to a}\sqrt[n] {f(x)}=\sqrt[n] {\lim\limits_{x \to a}f(x)}$ Note, $\underline n$ is a positive integer. If $n$ is even, we assume that $\lim\limits_{x \to a}f(x) > 0$
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