University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Section 2.5 - Continuity - Exercises - Page 95: 38


$$\lim_{x\to1}\cos^{-1}(\ln\sqrt x)=\frac{\pi}{2}$$ The function is continuous at $x=1$.

Work Step by Step

*Recall Theorem 10: If $g$ is continuous at $b$ and $\lim_{x\to c}f(x)=b$, then $$\lim_{x\to c}g(f(x))=g(b)=g(\lim_{x\to c}f(x))$$ $$A=\lim_{x\to1}f(x)=\lim_{x\to1}\cos^{-1}(\ln\sqrt x)$$ Apply Theorem 10 here: $$A=\cos^{-1}(\ln\sqrt{\lim_{x\to1}x})$$ $$A=\cos^{-1}(\ln\sqrt1)=f(1)$$ $$A=\cos^{-1}(\ln1)$$ $$A=\cos^{-1}0$$ $$A=\frac{\pi}{2}$$ As shown above, $\lim_{x\to1}f(x)=f(1)$, so the function is continuous at $x=1$.
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