Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.6 Calculating Limits Using the Limit Laws - 1.6 Exercises - Page 70: 33

Answer

(a) Estimate the value of $\lim\limits_{x \to 0} \frac{x}{\sqrt {1+3x }- 1}$ by graphing the function b) Make a table of values of $f(x)$ for $x$ close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct.

Work Step by Step

$$\lim\limits_{x \to 0} \frac{x}{\sqrt {1+3x}-1}=\lim\limits_{x \to 0} \frac{x}{\sqrt {1+3x}-1} * \frac{\sqrt {1+3x}+1}{\sqrt {1+3x}+1}= \lim\limits_{x \to 0} \frac{x(\sqrt {1+3x}+1)}{(1+3x)-1} = \lim\limits_{x \to 0} \frac{\sqrt {1+3x}+1}{3} =\frac{\sqrt {1+3(0)}+1}{3} = \frac{2}{3} $$
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