Calculus: Early Transcendentals 8th Edition

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

Chapter 2 - Section 2.5 - Continuity - 2.5 Exercises: 16

Answer

Prove that $\lim\limits_{x\to a}g(x)=g(a)$ for $\forall a\in(-\infty,-2)$

Work Step by Step

*NOTES TO REMEMBER: $f(x)$ is continuous on the interval if and only if it is continuous at every point in the interval. In other words, $f(x)$ is continuous on the interval $(u,v)$ if and only if for $\forall a\in(u,v)$, we have $$\lim\limits_{x\to a}f(x)=f(a)$$ For $\forall a\in(-\infty,-2)$, we consider $\lim\limits_{x\to a}g(x)$ $=\lim\limits_{x\to a}\frac{x-1}{3x+6}$ $=\frac{\lim\limits_{x\to a}(x-1)}{\lim\limits_{x\to a}(3x+6)}$ $=\frac{\lim\limits_{x\to a}x-\lim\limits_{x\to a}1}{3\lim\limits_{x\to a}x+\lim\limits_{x\to a}6}$ $=\frac{a-1}{3a+6}$ $=g(a)$ Therefore, $g(x)$ is continuous on the interval $(-\infty,-2)$.
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