Answered: cos X if x + 2 f(x) = if x = 2 1

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter6: Fractions

Section6.1: Simplifying Fractions

Problem 38WE

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Question

Determine for what numbers, if any, the shown function is discontinuous. Construct a table to find any required limits.

Expert Solution

Step 1

Given:

$f (x) = \{\begin{cases} \frac{\cos x}{x - \frac{π}{2}}, if x \neq \frac{π}{2} \\ 1, if x = \frac{π}{2} \end{cases}$

To Find:

Point of Discontinuity.

Concept:

$f (x) is continuous at x = a if : (i) . \lim_{x \to a} f (x) exist and finite (i i) . \lim_{x \to a} f (x) = f (a)$

Step 2

Explanation:

$f (x) = \{\begin{cases} \frac{\cos x}{x - \frac{π}{2}}, if x \neq \frac{π}{2} \\ 1, if x = \frac{π}{2} \end{cases} We will check the continuity of f (x) at x = \frac{π}{2} \lim_{x \to \frac{π}{2}} f (x) = \lim_{x \to \frac{π}{2}} \frac{\cos x}{x - \frac{π}{2}} = \frac{0}{0} Now, applying L-Hopital's Rule: \Rightarrow \lim_{x \to \frac{π}{2}} \frac{\cos x}{x - \frac{π}{2}} = \lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} (\cos x)}{\frac{d}{d x} (x - \frac{π}{2})} = \lim_{x \to \frac{π}{2}} \frac{- \sin x}{1} = \lim_{x \to \frac{π}{2}} - \sin x = - 1 \Rightarrow \lim_{x \to \frac{π}{2}} \frac{\cos x}{x - \frac{π}{2}} = - 1 . . . . . (i) ∵ f (\frac{π}{2}) = 1 . . . . . (i i) ∴ from (i) and (i i), we get \Rightarrow \lim_{x \to \frac{π}{2}} f (x) \neq f (\frac{π}{2}) Hence, f (x) is discontinuous at x = \frac{π}{2} .$

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