lect11_2

# lect11_2 - Calculus of Vector-Valued Functions(11.2 1...

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Unformatted text preview: Calculus of Vector-Valued Functions - (11.2) 1. Limits: Let r ! ! t " "# f ! t " , g ! t " , h ! t " \$ be a vector-valued function. Then lim t % a r ! ! t " " lim t % a # f ! t " , g ! t " , h ! t " \$"# lim t % a f ! t " , lim t % a g ! t " , lim t % a h ! t " \$ provided lim t % a f ! t " lim t % a g ! t " lim t % a h ! t " all exist. If one of the limits does not exists, then lim t % a r ! ! t " does not exist. Example Find the limit if it exists. a . lim t % # 1 t 2 & 1 , tant , ln ! t & 1 " \$ b . lim t % ! # 1 " 2 t 2 3 t 2 & t , te " 2 t , sin ! ! t " \$ c . lim t " # sint t , t lnt , 1 t " 1 \$ a. lim t % # 1 t 2 & 1 , tan t , ln ! t & 1 " \$"# t , 0, 0 \$ b. lim t % ! # 1 " 2 t 2 3 t 2 & t , te " 2 t , sin ! ! t " \$"# " 2 3 , 0, DNE \$ , so the limit does not exist. c. lim t " # sin t t , t ln t , 1 t " 1 \$"# 1, "! " 0, " 1 \$"# 1,0, " 1 \$ 2. Continuity The vector-valued function r ! ! t " "# f ! t " , g ! t " , h ! t " \$ is continuous at t " a whenever lim t % a r ! ! t " " r ! ! a " . So r ! ! t " is continuous at t " a if and only if lim t % a f ! t " " f ! a " lim t % a g ! t " " g ! a " lim t % a h ! t " " h ! a " . Again if one them is not continuous at t " a , r ! ! t " is not continuous at t " a . Example Determine for what values of t the vector-valued function r " ! t " "# tant , ln ! t & 1 " , 1 " t \$ is continuous. f ! t " " tan ! t " , D f " t : " 3 ! 2 , " ! 2 , " !...
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## This note was uploaded on 02/05/2012 for the course MATH 2142 taught by Professor Lerna during the Fall '10 term at FIU.

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lect11_2 - Calculus of Vector-Valued Functions(11.2 1...

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