Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.4 Limits and Continuity - Exercises - Page 67: 61


a) No value of $g(1)$; b) $g(1)=-\dfrac{\pi}{2}$

Work Step by Step

We are given the function: $g(x)=\tan^{-1}\left(\dfrac{1}{x-1}\right)$, $x\not=1$ a) Graph the function: The function has a jump discontinuity at $x=0$, therefore we cannot make it continuous no matter which value $g(1)$ would take. b) In order to make the function left-continuous in $x=1$, we should have: $g(1)=\displaystyle\lim_x\rightarrow 1^{-} \tan^{-1}\left(\dfrac{1}{x-1}\right)=-\dfrac{\pi}{2}$ Therefore we must have:\\ $g(1)=-\dfrac{\pi}{2}$
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