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Advanced Calculus
¾Í´ÕÇèÒ¼Áà¾Ôè§àÃÔèÁàÃÕ¹ Ad cal ¹èФÃѺàÅÂÍÂÒ¡Êͺ¶ÒÁÇèÒ·Óà຺¹Õéä´éÁÑé¹èФÃѺ
Proof that If $\displaystyle \lim_{P\rightarrow A} f(P)=L$ and $\displaystyle \lim_{p\rightarrow A} g(P)=M$ then $\displaystyle \lim_{P\rightarrow A} f(P)g(P)=L\cdot M$ ¨Ò¡ $\displaystyle \lim_{P\rightarrow A} f(P)=L$ ààÅÐ $\displaystyle \lim_{p\rightarrow A} g(P)=M$ ä´éÇèÒ $\displaystyle \forall \epsilon>0 $ ¨ÐÁÕ $\delta_1,\delta_2 >0$ ·Õè«Öè§ $0<||P-A||<\delta_1,\delta_2$ ·Õè·ÓãËéà¡Ô´ $|f(P)-L|<\dfrac{-(|L|+|M|)+\sqrt{(|L|+|M|)^2+4\epsilon}}{2}$ ààÅÐ $|g(P)-M|<\dfrac{-(|L|+|M|)+\sqrt{(|L|+|M|)^2+4\epsilon}}{2}$ µÒÁÅӴѺ NOTE $\displaystyle\epsilon_0=\dfrac{-(|L|+|M|)+\sqrt{(|L|+|M|)^2+4\epsilon}}{2}$ ¾Ô¨ÒÃ³Ò $\displaystyle |f(P)g(P)-L\cdot M|=|(f(P)-L)(g(P)-M)+L(g(P)-M)+M(f(P)-L)|$ $\displaystyle \le |f(P)-L||g(P)-M|+|L||g(P)-M|+|M||f(P)-L|\le \epsilon_0^2+(|L|+|M|)\epsilon_0=\epsilon$
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ÍÕ¡Íѹà»ç¹¡ÒÃËÒùФÃѺ
Lemma If $L\not=0$ and $\displaystyle \lim_{P\rightarrow A} f(P)=L$ then $\displaystyle \lim_{P\rightarrow A}\frac{1}{f(P)}=\frac{1}{L}$ Proof: $\forall \epsilon>0$ there exist $\delta>0$ «Öè§ $0<||P-A||<\delta$ ·ÓãËéà¡Ô´ $|f(P)-L|<t\epsilon$ àÁ×èÍ $t\in\mathbb{R^+}$ «Öè§ $t<|Lf(P)|$ ¾Ô¨ÒÃ³Ò $$\Big|\frac{1}{f(P)}-\frac{1}{L}\Big|=\Big|\frac{f(P)-L}{Lf(P)}\Big|<\frac{t\epsilon}{|Lf(P)|}<\frac{t\epsilon}{t}=\epsilon$$ ´Ñ§¹Ñ鹨ҡ¼Å¡Òäٳ $\displaystyle\lim_{P\rightarrow A} \frac{f(P)}{g(P)}=(\lim_{P\rightarrow A} f(P))\Big(\lim_{P\rightarrow A}\frac{1}{g(P)}\Big)=\frac{L}{M}$ µÒÁµéͧ¡ÒÃ
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Vouloir c'est pouvoir |
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äÁèµéͧÍÑ´ε¢¹Ò´¹Ñ鹡çä´éÁÑé§àÃÒÁÕ ãªé·ÄÉ®Õà¡èÒæẺsequential criterion for functional limit +algebraic limit thm for sequences ¡ç¹èÒ¨ÐÍÍ¡äÁèÂÒ¡
áµè¶éҨкÙê¨ÃÔ§æ¢éÍáá Áѹ¹èҨФÅÕ¹ä´é¡ÇèÒ¹Õé Åͧlet ε>0 be arbitrary. ·Õ¹ÕéàÅ×Í¡δ=Max{δ_1,δ_2} áÅéǾÂÒÂÒÁàÍÒÍÊÁ¡ÒÃÍѹà¡èÒ2ÍѹÁÒ bondÍѹãËÁè
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Mathematics, rightly viewed possesses not only truth, but supreme beauty. B.R. |
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¾Ç¡¤èÒà©ÅÕè AM. GM. HM. ¹Õèµéͧ¤Åèͧ˹è͹ФÃѺ äÁè§Ñé¹Íèҹ˹ѧÊ×ÍäÁèÃÙéàÃ×èͧ ÊÁѼÁàÃÕ¹áÃ¡æ ¡ç»ÃзѺã¨ÍÒ¨ÒÃÂì µÃ§·Õè·èÒ¹¼Ù¡ÊÔ觵èÒ§æ ·Ò§¤³ÔµÈÒʵÃìà¢éÒäÇé´éÇ¡ѹà»ç¹ÊÁ¡Òà ÁÕ¢Ñ鹵͹ÇÕ¸Õ¤Ô´ ·èÒ¹¤ÅèͧÁÒ¡àÅÂ
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