Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.2 Limits: A Numerical and Graphical Approach - Exercises - Page 54: 34


$\displaystyle\lim_{x\rightarrow 1^{+}} \dfrac{\sec^{-1} x}{\sqrt{x-1}}\approx 1.4142$

Work Step by Step

We have to estimate the limit: $\displaystyle\lim_{x\rightarrow 1^{+}} \dfrac{\sec^{-1} x}{\sqrt{x-1}}$ Compute $\dfrac{\sec^{-1} x}{\sqrt{x-1}}$ for values of $x$ close to 1, approaching from the right: $\dfrac{\sec^{-1} 2}{\sqrt{2-1}}\approx 1.0471976$ $\dfrac{\sec^{-1} 1.5}{\sqrt{1.5-1}}\approx 1.1894507$ $\dfrac{\sec^{-1} 1.1}{\sqrt{1.1-1}}\approx 1.3588297$ $\dfrac{\sec^{-1} 1.01}{\sqrt{1.01-1}}\approx 1.4083587$ $\dfrac{\sec^{-1} 1.001}{\sqrt{1.001-1}}\approx 1.4136247$ $\dfrac{\sec^{-1} 1.0001}{\sqrt{1.0001-1}}\approx 1.4141546$ $\dfrac{\sec^{-1} 1.00001}{\sqrt{1.00001-1}}\approx 1.4142077$ $\dfrac{\sec^{-1} 1.000001}{\sqrt{1.000001-1}}\approx 1.414213$ Therefore we got: $\displaystyle\lim_{x\rightarrow 1^{+}} \dfrac{\sec^{-1} x}{\sqrt{x-1}}\approx 1.4142$
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