09.29 - Section 1.5: Continuity (continued) Theorem 1.5.7:...

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(continued) Theorem 1.5.7: If lim x a g ( x ) = b and lim x b f ( x ) = c , then lim x a f ( g ( x )) = c , provided that the function f is continuous at b . Note that in the statement of the theorem, we could have written “ f ( b ) = c ” instead of “lim x b f ( x ) = c ” as our second hypothesis, since our third hypothesis (the continuity of f at b ) implies that f ( b ) and lim x b f ( x ) are equal. Application: Evaluate lim x 0 cos(sin( x )). Answer: Put f ( x ) = cos x , g ( x ) = sin x , a = 0, and b = 0. Since lim x a g ( x ) = lim x 0 sin x = sin 0 = 0 = b (by the continuity of trigonometric functions), and since lim x b f ( x ) = lim x 0 cos x = cos 0 = 1 = c (again by the continuity of trigonometric functions), and since f ( x ) = cos x is continuous at b = 0 (yet again by the continuity of trigonometric functions), all three required hypothesis of Theorem 1.5.7 are satisfied, so lim x 0 cos(sin( x )) = lim x
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This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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09.29 - Section 1.5: Continuity (continued) Theorem 1.5.7:...

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