我們的微積分用的是非常怪異的Calculus for biology and medicine 我看了以後覺得寫得好爛 好像重要的東西全部都變成optional 後面還出現一堆矩陣和集合 真是神奇的東西 從開學到現在才剛剛教完A本那些基本的極限證明 可是啊 我看書上的証法和上課教的好像不太一樣耶 寫上課教的會不會被扣分啊 <課本寫的> Use the formal definition of limits to show that lim√x=2 x→4 Sol: We assume throughout that x>0.We need to show that for every ε>0 there corresponds a numberδ>0 so that ∣√x-2∣<ε whenever 0<∣x-4∣<δ. Now ∣√x-2∣<ε is equivalent to -ε<√x-2<ε 2-ε<√x<2+ε To square all sides,we need to make sure that 2-ε≧0 and 2+ε≧0 Since ε>0 by assumption,2-ε≧0 requires us to assume 0<ε≦2.With 0<ε≦2, squaring yields (2-ε)^2<x<(2+ε)^2 4-4ε+ε^2<x<4+4ε+ε^2 Since 4-4ε+ε^2 > 4-4ε-ε^2,we have 4-(4ε+ε^2)< x< 4+(4ε+ε^2) or ∣x-4∣<4ε+ε^2 This suggests that if we set δ=4ε+ε^2 when 0<ε≦2,then 0<∣x-4∣<δ implies ∣√x-2∣<ε.This is indeed the case,namely if 0<∣x-4∣<4ε+ε^2, then -4ε-ε^2 <x-4< 4ε+ε^2, or 4-4ε-ε^2 <x-4< 4+4ε+ε^2. This shows that (2-ε)^2<x<(2+ε)^2.Taking square roots(and observing that x>0) 2-ε<√x<2+ε or -ε<√x-2<ε,which is the same as ∣√x-2∣<ε. When ε>2,we can choose δ=12(or any other smaller value).The value δ=12 comes from pluggingε=2 into δ=4ε+ε^2.With δ=12,if 0<∣x-4∣<12,then -12<x-4<12 or -8<x<16 .Since x>0, we have 0<x<16,or after taking square roots, 0<√x<4,which is the same as -2<√x-2<2 or ∣√x-2∣<2<ε. Orz,可以只寫上課教的那個嗎??? 怎麼覺得他廢話一堆啊 --

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