Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.5 Derivatives of Trigonometric Functions - 3.5 Exercises: 52

Answer

\[ = - 1\]

Work Step by Step

\[\begin{gathered} \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\,\,\,\frac{{\cos x}}{{x - \frac{\pi }{2}}}\,\, \hfill \\ \hfill \\ let\,,\,\,\,\cos x = \sin \,\left( {\frac{\pi }{2} - 2} \right) \hfill \\ \hfill \\ = - \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\sin \,\left( {\frac{\pi }{2} - x} \right)}}{{\frac{\pi }{2} - x}} \hfill \\ \hfill \\ using\,\,special\,\,\,limit:\,\,\,\mathop {\lim }\limits_{x \to 0} \,\,\frac{{\sin x}}{x} = 1 \hfill \\ then \hfill \\ \hfill \\ - \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\sin \,\left( {\frac{\pi }{2} - x} \right)}}{{\frac{\pi }{2} - x}} = - 1 \hfill \\ \end{gathered} \]
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