Եͧѧѹ

Ԫ ԵʵԵͧѧѹ ǤԴ鹰ҹͧ Ե (ҤɮբͧŤ)

Ҿٴ ѧѹ f Ե L ش p ¤ Ѿͧ f L شش p Ѻҧ繷ҧù աá˹鹤á ǧ¢ͧ ʵȵɷ 19

ѧѹԧзҧ

˹ f: (M,dM) -> (N,dN) 繡觤ҧ (繿ѧѹ) ԧзҧ ͧ, С˹ pM LN, ҨС 'Եͧ f p L' ¹

 \lim_{x \to p}f(x) = L

ѺءҢͧ ε > 0

δ > 0 Ѻء xM dM(x, p) < δ , dN(f(x), L) < ε

ѧѹҨԧ

ͧӹǹԧ鹨ӹǹԧ ·öͧ繻ԧзҧ d(x,y): = | xy | . ǡѺ 鹨ӹǹԧ (鹨ӹǹԧ +∞ -∞ 仴) öͧ繻ԧзҧ d(x,y): = | arctan(x) − arctan(y) |

ԵͧѧѹҨԧش㴨ش˹

f 繿ѧѹҨԧ Ҩ¹

 \lim_{x \to p}f(x) = L 
ѺءҢͧ ε > 0 (Ҩ) еͧ δ > 0 ҧ˹觤 ѺءҢͧӹǹԧ x 0 < |x-p| < δ, |f(x)-L| < ε
 \lim_{x \to p}f(x) = \infty
ѺءҢͧ R > 0 (Ҩ˭) еͧ δ > 0 ҧ˹觤 ѺءҢͧӹǹԧ x 0 < |x-p| < δ, f(x) > R;
 \lim_{x \to p}f(x) = -\infty
ѺءҢͧ R < 0 еͧ δ > 0 ҧ˹觤 ѺءҢͧӹǹԧ x 0 < |x-p| < δ, f(x) < R.
ԵͧѧѹҨԧ ͹ѹ
Եͧѧѹ ͹ѹ Ѻ ε > 0 S > 0 ҧ˹觤 |f(x)-L| < ε Ѻ x > S

f(x) 繿ѧѹҨԧ ҨоԨóԵͧѧѹ x Ŵŧҧշش

Ҩ¹

 \lim_{x \to \infty}f(x) = L


ѧѹԧ͹

йҺԧ͹ յѴ (metric) d(x,y): = | xy | 繻ԧзҧ (metric space) 蹡ѹ ԵͧҾٴ֧ѧѹԧ͹


Եͧѧѹش㴨ش˹


f 繿ѧѹԧ͹ Ҩ¹

 \lim_{x \to p}f(x) = L

Ѻ ε > 0 δ >0 ҧ 1 Ѻӹǹԧ x 0<|x-p|<δ |f(x)-L|<ε

繡óվɢͧѧѹԧзҧշ M N йҺԧ͹




Եͧѧѹ ͹ѹ


Ҩ¹

 \lim_{x \to \infty}f(x) = L

Ѻ ε > 0 S >0 Ѻӹǹԧ͹ |x|>S Ҩ |f(x)-L|<ε

ѧѹҨԧ

\lim_{x \to 3}x^2=9
Եͧ x2 x 3 9 㹡óչ ѧѹ鹵ͧ ФҢͧѹչش Ե֧ҡѺ᷹ҿѧѹ




\lim_{x \to 0^+}x^x=1


Եͧ xx x 0 ҡҧҤ 1


ѧѹԧзҧ
  • z 繨ӹǹԧ͹ · |z| < 1 ӴѺ z, z2, z3, ... ͧӹǹԧ͹Ե 0 âҤԵ ӹǹҹ '¹繡' شԴ 鹡͡Է

  • 㹻ԧзҧ C[a,b] ͧѧѹͧ ǧ [a,b] зҧ鹨ҡ Supremum norm Ҫԡءö¹ٻͧԵͧӴѺͧ ѧѹع Ңͧ ɮպ⵹-ʵ (Stone-Weierstrass theorem)

觷Ңͧ http://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%B4%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99




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