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Unformatted text preview: Math 21A Brian Osserman Practice Exam 1 Answer Key 1 (32 pts.) Determine whether or not the following limits exist, and calculate them. If the limit does not exist as a number, state whether or not it can be written as ∞ or∞ . (a) lim x → 3 x 3 x 2 2 x 3 Answer: lim x → 3 x 3 x 2 2 x 3 = 1 / 4 (b) lim t → √ t +1 1 t Answer: lim t → √ t +1 1 t = 1 / 2 (c) lim x → 1 + 2 x 2 1 x Answer: lim x → 1 + 2 x 2 1 x =∞ (d) lim x →∞ sin x Answer: lim x →∞ sin x does not exist 2 (8 pts.) Using the sandwich theorem, show that lim x → x 4 (1 cos x ) = 0 . Answer: ≤ x 4 (1 cos x ) ≤ 2 x 4 . 3 (12 pts.) Directly from the de nition of a limit, show that lim x → 3 x 2 = 9 . Answer: Given > , if < 9 then we can use δ = min { 3 √ 9 , √ 9 + 3 } . (Notice that if you need to know a precise valid formula for δ , you might need to gure out which possibility is smaller. But to prove the limit is 9 , it is enough to know that a δ exists, so taking the minimum of two possibilities...
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This homework help was uploaded on 04/07/2008 for the course MATH 21A taught by Professor Osserman during the Fall '07 term at UC Davis.
 Fall '07
 Osserman
 Limits

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