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Unformatted text preview: Math 1110 Practice Prelim 3 Solutions 1. Evaluate the following limits if they exist. If a limit does not exist, explain why. (a) lim θ → (1 + θ ) csc( θ ) (b) lim x → 1 x 3 (c) lim x → 3 x 2 x 12 x +3 Solution: (a) Writing L = lim θ → (1 + θ ) csc( θ ) , we have ln( L ) = lim θ → ln((1 + θ ) csc( θ ) ) = lim θ → ln(1 + θ ) sin( θ ) . This is a 0 / 0 indeterminate form, so l’Hˆ opital’s Rule applies. Therefore ln( L ) = lim θ → 1 1+ θ cos( θ ) = 1 . So lim θ → (1 + θ ) csc( θ ) = L = e 1 = e . (b) As x approaches 0, the denominator approaches 0, while the numerator stays constant at 1. So the limit does not exist. (c) Using l’Hˆ opital’s Rule, we see that the answer is 7. 2. Evaluate the following expressions. (a) R π cos( x ) dx (b) R 2 x x 1 t 2 dt (c) d dx R x 2 a sin( t ) dt (d) R (sin(2 θ )) e sin 2 θ dθ Solution: (a) 0 (b) x 7 x 3 3 (c) 2 x sin( x 2 ) (d) Let u = sin 2 θ so that du = 2sin θ cos θ = sin(2 θ ). Then R (sin(2...
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This note was uploaded on 12/06/2010 for the course MATH 1110 taught by Professor Martin,c. during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 MARTIN,C.
 Calculus, Limits

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