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Unformatted text preview: , and ln( x ) . 4 . (a) Show that sin( x ) ≃ x is a good approximation for small x by using Taylor’s theorem to obtain  sin( x ) − x  ≤ 6 1  x  3 ∀ x ∈ R . (b) Use (a) to calculate the limit for different values of a ∈ R and c > of the function x a sin(  x  − c ) (from Rudin pg.115 #13) as x → ∞ . 5 . (a) Assume f : (0 , 1] −→ R is differentiable and  f ′ ( x )  ≤ M for all x ∈ (0 , 1] . Deﬁne the sequence a n = f (1 /n ) and prove that a n converges. (b) Problem # 26 page 119 in Rudin . MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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This document was uploaded on 10/26/2011 for the course MATH 18.100B at MIT.
 Fall '10
 Prof.KatrinWehrheim
 Mean Value Theorem

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