엡실론_델타_예제_01 (rev. 1.7)
Up: 극한(limit)
김도형 ¶
$\displaystyle \lim_{x\to3}(4x-5)=7$ 을 증명하라.
pf.
We want: $\displaystyle \forall \epsilon>0,\,\exists \delta >0$
pf.
We want: $\displaystyle \forall \epsilon>0,\,\exists \delta >0$
s.t. $\displaystyle 0<|x-3|\delta\Rightarrow |(4x-5)-7|<\epsilon$
Delta Epsilon Proofs.pdf 1 ¶
$\displaystyle \lim_{x\to 2}(3x-1)=5$
다시 말해
Find an $\displaystyle \epsilon>0$ such that if $\displaystyle 0<|x-2|<\delta$ then $\displaystyle |(3x-1)-5|<\epsilon$
다시 말해
Find an $\displaystyle \epsilon>0$ such that if $\displaystyle 0<|x-2|<\delta$ then $\displaystyle |(3x-1)-5|<\epsilon$
먼저 delta를 찾는다
$\displaystyle |3x-6|<\epsilon$
$\displaystyle |x-2|<\epsilon/3$
$\displaystyle |3x-6|<\epsilon$
$\displaystyle |x-2|<\epsilon/3$
그래서 $\displaystyle \delta=\frac{\epsilon}3$ 으로 pick, 그러면 $\displaystyle |x-2|<\frac{\epsilon}{3}$ 이 되는지?
Yes.
Delta Epsilon Proofs.pdf 2 ¶
$\displaystyle \lim_{x\to 2}(3x^2-4x+1)=5$ 를 증명하라.
다시 말해 양수 epsilon을 찾아라. such that
if $\displaystyle 0<|x-2|<\delta$ then $\displaystyle |(3x^2-4x+1)-5|<\epsilon$
if $\displaystyle 0<|x-2|<\delta$ then $\displaystyle |(3x^2-4x+1)-5|<\epsilon$
$\displaystyle 3x^2-4x-4=(x-2)(3x+2)$
TODO
이상 http://mathnmath.tistory.com/36 의
Delta Epsilon Proofs.pdf 참조했음
"This handout is available at: www.uvu.edu/mathlab/handouts.html"
Delta Epsilon Proofs.pdf 참조했음
"This handout is available at: www.uvu.edu/mathlab/handouts.html"