{"id":4482,"date":"2020-05-13T12:32:20","date_gmt":"2020-05-13T03:32:20","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4482"},"modified":"2020-09-14T13:08:43","modified_gmt":"2020-09-14T04:08:43","slug":"calculus-exercises-on-the-limits-of-a-multivariable-functions-solution","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-exercises-on-the-limits-of-a-multivariable-functions-solution\/","title":{"rendered":"\ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uadf9\ud55c \uc99d\uba85 \uc5f0\uc2b5\ubb38\uc81c \ud574\uc124"},"content":{"rendered":"<style type=\"text\/css\">\ndiv.problemquestion {\n\tmargin-bottom: 2em;\n}\ndiv.problemquestion p:last-child {\n\tmargin-bottom: 0em;\n}\n<\/style>\n<p><!-- div class=\"problemquestion\">\n\n\n<p>\ucd94\uac00 \ubb38\uc81c\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \ubb38\uc81c\uc758 \uc99d\uba85\uc740 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc758 \uc5c4\ubc00\ud55c \uc815\uc758(\\(\\epsilon - \\delta\\) \ub17c\ubc95)\ub97c \uc0ac\uc6a9\ud588\uc2b5\ub2c8\ub2e4.<\/p>\n\n\n<\/div -->\n<div class=\"box\">\n<p><span class=\"definition\">\ubb38\uc81c 1.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) = \\sin x + \\sin y\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \ubaa8\ub4e0 \uc810 \\((x,\\,y)\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (\\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95\uc744 \uc0ac\uc6a9\ud560 \uac83.)<\/p>\n<\/div>\n<div class=\"solution\">\n<p><span class=\"definition\">\ud480\uc774.<\/span> \uc810 \\((x,\\,y)\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[\\delta = \\frac{\\epsilon}{2}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uba74 \ubaa8\ub4e0 \uc2e4\uc218 \\(s,\\) \\(t\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n\\lvert s-x \\rvert < \\delta \\quad &#038;\\Rightarrow \\quad \\lvert \\sin s - \\sin x \\rvert \\le \\lvert s-x \\rvert < \\delta = \\frac{\\epsilon}{2} , \\tag{1.1} \\\\[8pt]\n\\lvert t-y \\rvert < \\delta \\quad &#038;\\Rightarrow \\quad \\lvert \\sin t - \\sin y \\rvert \\le \\lvert t-y \\rvert < \\delta = \\frac{\\epsilon}{2} . \\tag{1.2}\n\\end{align}\n\\]\n\\(\\lvert (s,\\,t) - (x,\\,y) \\rvert < \\delta\\)\uc77c \ub54c\n\\[\\begin{align}\n\\lvert s-x \\vert = \\sqrt{ (s-x)^2 } \\le \\sqrt{ (s-x)^2 + (t-y)^2 } < \\delta , \\\\[8pt]\n\\lvert t-y \\vert = \\sqrt{ (t-y)^2 } \\le \\sqrt{ (s-x)^2 + (t-y)^2 } < \\delta  \n\\end{align}\\]\n\uc774\ubbc0\ub85c (1.1), (1.2)\uc5d0 \uc758\ud558\uc5ec\n\\[\n\\lvert \\sin s - \\sin x \\rvert < \\frac{\\epsilon}{2}, \\\\[6pt]\n\\lvert \\sin t - \\sin y \\rvert < \\frac{\\epsilon}{2}\n\\]\n\uc774\uba70, \ub530\ub77c\uc11c\n\\[\n\\begin{align}\n\\lvert f(s,\\,t) - f(x,\\,y) \\rvert \n&#038;= \\lvert (\\sin s + \\sin t) - (\\sin x + \\sin y) \\rvert \\\\[8pt]\n&#038;= \\lvert (\\sin s - \\sin x) + (\\sin t - \\sin y) \\rvert \\\\[6pt]\n&#038;\\le \\lvert \\sin s - \\sin x \\rvert + \\lvert \\sin t - \\sin y \\rvert < \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\n\\end{align}\n\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \\(f\\)\ub294 \\((x,\\,y)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \\((x,\\,y)\\)\uac00 \\(\\mathbb{R}^2\\)\uc758 \uc784\uc758\uc758 \uc810\uc774\ubbc0\ub85c, \\(f\\)\ub294 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### --><\/p>\n<div class=\"box\">\n<p><span class=\"definition\">\ubb38\uc81c 2.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\n1 \\quad &#038; \\text{if} \\,\\, y=x^2 \\text{ and } x > 0 \\\\[6pt]<br \/>\n0 \\quad &#038; \\text{otherwise}<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (\\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95\uc744 \uc0ac\uc6a9\ud560 \uac83.)<\/p>\n<\/div>\n<div class=\"solution\">\n<p><span class=\"definition\">\ud480\uc774.<\/span><br \/>\n\uc5f0\uc18d\uc758 \uc815\uc758\uc758 \ubd80\uc815\uc744 \uc99d\uba85\ud558\uc790. \uc989 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790.<br \/>\n\\[ \\exists \\epsilon > 0 \\,\\, \\forall \\delta > 0 \\,\\, \\exists ( x,\\,y) \\,\\,:\\,\\, \\left( \\lvert (x,\\,y) &#8211; (0,\\,0) \\rvert < \\delta \\,\\,\\wedge\\,\\, \\lvert f(x,\\,y) - f(0,\\,0) \\rvert \\ge \\epsilon \\right)\\]\n\uc774\uac83\uc744 \uc99d\uba85\ud558\uae30 \uc704\ud558\uc5ec \\(\\epsilon = 1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\delta > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\uc81c \\((0,\\,0)\\)\uacfc\uc758 \uac70\ub9ac\uac00 \\(\\delta\\)\ubcf4\ub2e4 \uc791\uc73c\uba74\uc11c \\(\\lvert f(x,\\,y) &#8211; f(0,\\,0)\\rvert \\ge \\epsilon\\)\uc778 \uc810 \\((x,\\,y)\\)\uac00 \uc874\uc7ac\ud568\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<br \/>\n\\[x = \\min \\left\\{ \\frac{1}{2} ,\\, \\frac{\\delta}{2} \\right\\},\\,\\, y=x^2 \\tag{2.1}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\delta \\le 1\\)\uc774\uba74<br \/>\n\\[x^2 + y^2 = \\frac{\\delta^2}{4} + \\frac{\\delta^4}{16} < \\frac{\\delta^2}{4} + \\frac{\\delta^2}{4} = \\frac{\\delta^2}{2}\\]\n\uc774\ubbc0\ub85c\n\\[\\lvert (x,\\,y) - (0,\\,0) \\rvert = \\sqrt{x^2 + y^2} < \\frac{\\delta}{\\sqrt{2}} < \\delta \\tag{2.2}\\]\n\uc774\uba70, \ub9cc\uc57d \\(\\delta > 1\\)\uc774\uba74<br \/>\n\\[x^2 + y^2 = \\left( \\frac{1}{2} \\right)^2 + \\left( \\frac{1}{2} \\right)^4 = \\frac{5}{16} < 1\\]\n\uc774\ubbc0\ub85c\n\\[\\lvert (x,\\,y) - (0,\\,0) \\rvert = \\sqrt{x^2 + y^2} < 1 < \\delta \\tag{2.3}\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (2.2), (2.3)\uc5d0 \uc758\ud558\uc5ec, \uc5b4\ub290 \uacbd\uc6b0\uc5d0\ub098 \\((x,\\,y)\\)\ub294 \\(\\lvert (x,\\,y) - (0,\\,0) \\rvert < \\delta\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7f0\ub370 (2.1)\uc5d0 \uc758\ud558\uc5ec \\((x,\\,y)\\)\ub294 \\(y=x^2\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c\n\\[\\lvert f(x,\\,y) - f(0,\\,0) \\rvert = f(x,\\,y) = 1 \\ge \\epsilon\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\((0,\\,0)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"definition\">\ucd94\uac00 \ubb38\uc81c.<\/span><br \/>\n\ubb38\uc81c 2\uc5d0\uc11c \uc815\uc758\ud55c \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec, \\((0,\\,0)\\)\uc5d0\uc11c \ubaa8\ub4e0 \ubc29\ud5a5\uc73c\ub85c\uc758 \\(f\\)\uc758 \ubc29\ud5a5\ubbf8\ubd84\uacc4\uc218\uac00 \\(0\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"solution\">\n<p><span class=\"definition\">\ud480\uc774.<\/span><br \/>\n\\(u = \\langle u_1 ,\\, u_2 \\rangle\\)\uac00 \ub2e8\uc704\ubca1\ud130\ub77c\uace0 \ud558\uc790. \\((D_u f)(0,\\,0) = 0\\)\uc744 \ubcf4\uc774\uae30 \uc704\ud574\uc11c\ub294 \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<br \/>\n\\[\\lim_{s\\to 0} \\frac{ f( 0+ su_1 ,\\, 0+ su_2 ) &#8211; f(0,\\,0) } {s} = 0.\\]<br \/>\n\\(f(0,\\,0) =0\\)\uc774\ubbc0\ub85c, \uc704 \ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\lim_{s\\to 0} \\frac{ f(  su_1 ,\\, su_2 )  } {s} = 0. \\tag{2.4}\\]<br \/>\n\ub9cc\uc57d \\(u_1 u_2 \\le 0\\)\uc774\ub77c\uba74 \\((su_1 ,\\, su_2 )\\)\ub294 \uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \uc81c 2 \uc0ac\ubd84\uba74, \uc81c 4 \uc0ac\ubd84\uba74 \ub610\ub294 \ucd95 \uc704\uc758 \uc810\uc774\ubbc0\ub85c \uace1\uc120<br \/>\n\\[y=x^2 ,\\,\\, x > 0 \\tag{2.5}\\]<br \/>\n\uc704\uc5d0 \ub193\uc5ec \uc788\uc9c0 \uc54a\ub2e4. \uc989 \uc774 \uacbd\uc6b0\uc5d0 \\(f(su_1 ,\\, su_2 ) =0\\)\uc774\ubbc0\ub85c (2.4)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(u_1 u_2 > 0\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \uc6d0\uc810\uc744 \uc9c0\ub098\uace0 \uae30\uc6b8\uae30\uac00 \uc591\uc218\uc778 \uc9c1\uc120\uc740 \uace1\uc120 (2.5)\uc640 \ub531 \ud55c \ubc88 \ub9cc\ub098\ubbc0\ub85c, \uc810 \\((su_1 ,\\, su_2 )\\)\ub294 \\(s\\)\ub97c \ubcc0\uc218\ub85c \uc0dd\uac01\ud588\uc744 \ub54c \uace1\uc120 (2.5)\uc640 \ub531 \ud55c \ubc88 \ub9cc\ub098\uba70, \ub9cc\ub098\ub294 \uc810\uc740 \uc81c 1 \uc0ac\ubd84\uba74\uc5d0 \uc788\ub2e4. \uadf8 \uad50\uc810\uc744 \\(P\\)\ub77c\uace0 \ud558\uace0, \\(P\\)\uc640 \uc6d0\uc810\uc758 \uac70\ub9ac\ub97c \\(\\delta\\)\ub77c\uace0 \ud558\uba74 \\(\\delta > 0\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(0 < \\lvert s-0 \\rvert < \\delta\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\((su_1 ,\\, su_2 )\\)\uc640 \\((0,\\,0)\\)\uc758 \uac70\ub9ac\ub294 \\(\\delta\\) \ubbf8\ub9cc\uc774\ubbc0\ub85c \\((su_1 ,\\, su_2 )\\)\ub294 \uace1\uc120 (2.5) \uc704\uc5d0 \ub193\uc5ec \uc788\uc9c0 \uc54a\ub2e4. \uadf8\ub7ec\ubbc0\ub85c\n\\[\\left\\lvert \\frac{f(su_1 ,\\, su_2 )}{s} -0 \\right\\rvert = \\left\\lvert \\frac{0}{s} \\right\\rvert = 0 < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (2.4)\uac00 \uc99d\uba85\ub418\uc5c8\ub2e4.\n\n\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### --><\/p>\n<div class=\"box\">\n<p><span class=\"definition\">\ubb38\uc81c 3.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\n\\frac{x^4 + y^4}{x^2 + y^2} \\quad &#038; \\text{if} \\,\\, (x,\\,y) \\ne (0,\\,0) \\\\[6pt]<br \/>\n0 \\quad &#038; \\text{if} \\,\\, (x,\\,y) = (0,\\,0)<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (\\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95\uc744 \uc0ac\uc6a9\ud560 \uac83.)<\/p>\n<\/div>\n<div class=\"solution\">\n<p><span class=\"definition\">\ud480\uc774.<\/span><br \/>\n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\sqrt{\\epsilon}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\[\\lvert (x,\\,y) &#8211; (0,\\,0) \\rvert < \\delta\\]\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\((x,\\,y) = (0,\\,0)\\)\uc774\uba74\n\\[\\lvert f(x,\\,y) - f(0,\\,0) \\rvert = 0 < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub9cc\uc57d \\((x,\\,y) \\ne (0,\\,0)\\)\uc774\uba74\n\\[\\begin{align}\n\\lvert f(x,\\,y) - f(0,\\,0) \\rvert \n&#038;= \\frac{x^4 + y^4}{x^2 + y^2} \\\\[6pt]\n&#038;\\le \\frac{x^4 + 2 x^2 y^2 + y^4}{x^2 + y^2} \\\\[4pt]\n&#038;= \\frac{ \\left( x^2 + y^2 \\right)^2}{x^2 + y^2} \\\\[6pt]\n&#038;= x^2 + y^2 < \\delta^2 = \\epsilon\n\\end{align}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\((0,\\,0)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### --><\/p>\n<div class=\"box\">\n<p><span class=\"definition\">\ubb38\uc81c 4.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\nx^2 + y^2 \\quad &#038; \\text{if} \\,\\, (x,\\,y) \\in \\mathbb{Q} \\times \\mathbb{Q} \\\\[6pt]<br \/>\n0 \\quad &#038; \\text{otherwise}<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c\ub9cc \uc5f0\uc18d\uc774\uace0 \ub2e4\ub978 \uc810\uc5d0\uc11c\ub294 \ubd88\uc5f0\uc18d\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624. (\\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95\uc744 \uc0ac\uc6a9\ud560 \uac83.)<\/p>\n<\/div>\n<div class=\"solution\">\n<p><span class=\"definition\">\ud480\uc774.<\/span><br \/>\n\uba3c\uc800 \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc790. \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\sqrt{\\epsilon}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[\\lvert (x,\\,y) &#8211; (0,\\,0) \\rvert < \\delta\\]\n\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\((x,\\,y) \\in \\mathbb{Q} \\times \\mathbb{Q}\\)\uc774\uba74\n\\[\\lvert f(x,\\,y) - f(0,\\,0) \\rvert = x^2 + y^2 = \\lvert (x,\\,y) - (0,\\,0) \\rvert ^2 < \\delta^2 = \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub9cc\uc57d \\((x,\\,y) \\notin \\mathbb{Q} \\times \\mathbb{Q}\\)\uc774\uba74\n\\[\\lvert f(x,\\,y) - f(0,\\,0) \\rvert = \\lvert 0-0 \\rvert =0 < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\((0,\\,0)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(f\\)\uac00 \uc6d0\uc810\uc774 \uc544\ub2cc \ub2e4\ub978 \uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc790. \uc6d0\uc810\uc774 \uc544\ub2cc \uc810 \\((a,\\,b)\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<\/p>\n<p>\ub9cc\uc57d \\((a,\\,b) \\in \\mathbb{Q} \\times \\mathbb{Q}\\)\uc774\uba74 \\(\\epsilon = f(a,\\,b)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\epsilon > 0\\)\uc774\ub2e4. \uc774\uc81c \\(\\delta > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubb34\ub9ac\uc218\uc758 \uc870\ubc00\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n0 &#038; < \\lvert x-a \\rvert < \\frac{\\delta}{2} , \\\\[6pt]\n0 &#038; < \\lvert y-b \\rvert < \\frac{\\delta}{2}\n\\end{align}\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub450 \ubb34\ub9ac\uc218 \\(x,\\) \\(y\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c\n\\[0 < \\lvert (x,\\,y) - (a,\\,b) \\rvert \\le \\lvert x-a \\rvert + \\lvert y-b \\rvert < \\delta \\tag{4.1}\\]\n\uac00 \uc131\ub9bd\ud558\uba70,\n\\[\\lvert f(x,\\,y) - f(a,\\,b) \\rvert = \\lvert 0- f(a,\\,b) \\rvert = f(a,\\,b) \\ge \\epsilon \\tag{4.2}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\((a,\\,b) \\notin \\mathbb{Q} \\times \\mathbb{Q}\\)\uc774\uba74 \\[\\epsilon = \\frac{a^2 + b^2}{4} \\]\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\epsilon > 0\\)\uc774\ub2e4. \ud45c\uae30\ub97c \ud3b8\ud558\uac8c \ud558\uae30 \uc704\ud558\uc5ec \\(\\delta_1 = \\frac{1}{2} \\lvert ( a,\\,b) \\rvert\\)\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(\\delta > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc720\ub9ac\uc218\uc758 \uc870\ubc00\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n0 &#038; < \\lvert x-a \\rvert < \\frac{1}{2} \\min \\left\\{ \\delta ,\\, \\delta_1 \\right\\} , \\\\[6pt]\n0 &#038; < \\lvert y-b \\rvert < \\frac{1}{2} \\min \\left\\{ \\delta ,\\, \\delta_1 \\right\\}\n\\end{align}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub450 \uc720\ub9ac\uc218 \\(x,\\) \\(y\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c\n\\[0 < \\lvert (x,\\,y) - (a,\\,b) \\rvert \\le \\lvert x-a \\rvert + \\lvert y-b \\rvert < \\min\\left\\{ \\delta ,\\, \\delta_1 \\right\\} \\tag{4.3}\\]\n\uc774 \uc131\ub9bd\ud558\uba70,\n\\[\\begin{align}\n\\lvert f(x,\\,y) - f(a,\\,b) \\rvert &#038;= x^2 + y^2\n= \\lvert (x,\\,y) \\rvert ^2 \\\\[6pt]\n&#038;= \\lvert (a,\\,b) + (x,\\,y) - (a,\\,b) \\rvert ^2 \\\\[6pt]\n&#038;\\ge \\left( \\lvert (a,\\,b) \\rvert - \\lvert (x,\\,y) - (a,\\,b) \\rvert \\right)^2 \\\\[6pt]\n&#038; = \\left( 2\\delta_1 - \\lvert (x,\\,y) - (a,\\,b) \\rvert \\right)^2 \\\\[6pt]\n&#038; > \\left( 2\\delta_1 &#8211; \\delta_1 \\right)^2 =  \\delta_1 {} ^2 \\\\[6pt]<br \/>\n&#038;= \\frac{a^2 + b^2}{4} = \\epsilon \\tag{4.4}<br \/>\n\\end{align}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p><p>(4.1), (4.2), (4.3), (4.4)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[0 < \\lvert (x,\\,y) - (a,\\,b) \\rvert < \\delta \\,\\,\\wedge\\,\\, \\lvert f(x,\\,y) -f(a,\\,b) \\rvert \\ge \\epsilon\\]\n\uc778 \uc810 \\((x,\\,y)\\)\uac00 \uc874\uc7ac\ud558\ubbc0\ub85c, \\(f\\)\ub294 \\((a,\\,b)\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub2e4.\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"definition\">\ucd94\uac00 \ubb38\uc81c.<\/span><br \/>\n\ubb38\uc81c 4\uc5d0\uc11c \uc815\uc758\ud55c \ud568\uc218 \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"solution\">\n<p><span class=\"definition\">\ud480\uc774.<\/span><br \/>\n\\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c \ud3b8\ubbf8\ubd84 \uac00\ub2a5\ud568\uc740 \uc790\uba85\ud558\ub2e4. \uc989<br \/>\n\\[\\frac{\\partial f}{\\partial x} (0,\\,0) = 0 ,\\,\\,\\, \\frac{\\partial f}{\\partial y} (0,\\,0) = 0 \\tag{4.5}\\]<br \/>\n\uc774\ub2e4. \uc774\uc81c \uc120\ud615\ubcc0\ud658 \\(A : \\mathbb{R}^2 \\,\\rightarrow \\, \\mathbb{R}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc740 \ud589\ub82c\ub85c \uc815\uc758\ud558\uc790.<br \/>\n\\[A = \\left[ 0 \\quad 0 \\right].\\]<br \/>\n\uadf8\ub7ec\uba74 \uc784\uc758\uc758 \ubca1\ud130 \\(\\textbf{h} = \\binom{h_1}{h_2}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[A \\textbf{h} = 0\\]<br \/>\n\uc774\ubbc0\ub85c \\(A\\)\ub294 \\(\\mathbb{R}^2\\)\uc758 \ubaa8\ub4e0 \uc810\uc744 \\(0\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \uc120\ud615\uc0ac\uc0c1\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(\\textbf{h} = \\langle h_1 ,\\, h_2 \\rangle\\)\uac00 \\(\\textbf{0}\\)\uc774 \uc544\ub2cc 2\ucc28\uc6d0 \ubca1\ud130\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\textbf{h} \\in \\mathbb{Q} \\times \\mathbb{Q}\\)\uc774\uba74<br \/>\n\\[f(\\textbf{h}) = h_1 {}^2 + h_2 {}^2 = \\lvert \\textbf{h} \\rvert^2 \\]<br \/>\n\uc774\uace0, \ub9cc\uc57d \\(\\textbf{h} \\notin \\mathbb{Q} \\times \\mathbb{Q}\\)\uc774\uba74<br \/>\n\\[f(\\textbf{h}) = 0 \\le \\lvert \\textbf{h} \\rvert^2\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5b4\ub290\ub54c\ub098<br \/>\n\\[\\lvert f(\\textbf{h}) \\rvert \\le \\lvert \\textbf{h} \\rvert^2 \\tag{4.6}\\]<br \/>\n\uc774\ub2e4. \uc774\uc81c \ub2e4\uc74c \uadf9\ud55c\uc744 \uacc4\uc0b0\ud558\uc790.<br \/>\n\\[\\lim_{\\textbf{h} \\to \\textbf{0}} \\frac{\\lvert f(\\textbf{0}+\\textbf{h}) &#8211; f(\\textbf{0}) &#8211; A\\textbf{h} \\rvert}{\\lvert \\textbf{h} \\rvert}.\\tag{4.7}\\]<br \/>\n\\(f(\\textbf{0})=0\\)\uc774\uace0 \\(A\\textbf{h} = 0\\)\uc774\ubbc0\ub85c \uc704 \uadf9\ud55c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\lim_{\\textbf{h} \\to \\textbf{0}} \\frac{\\lvert f(\\textbf{h}) \\rvert}{\\lvert \\textbf{h} \\rvert}.\\tag{4.8}\\]<br \/>\n\uadf8\ub7f0\ub370 (4.6)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\frac{\\lvert f(\\textbf{h}) \\rvert}{\\lvert \\textbf{h} \\rvert} \\le \\frac{\\lvert \\textbf{h} \\rvert^2}{\\lvert \\textbf{h} \\rvert} = \\lvert \\textbf{h} \\rvert\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\lvert \\textbf{h} \\rvert \\,\\rightarrow\\, \\textbf{0}\\)\uc77c \ub54c<br \/>\n\\[\\frac{\\lvert f(\\textbf{h}) \\rvert}{\\lvert \\textbf{h} \\rvert} \\,\\rightarrow\\,0\\]<br \/>\n\uc774\ub2e4. \uc989 (4.7)\uc758 \uadf9\ud55c\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \\(f\\)\ub294 \\((0,\\,0)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \uadf8 \uc810\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub294 \ud589\ub82c \\(A\\)\ub85c \ud45c\ud604\ub418\ub294 \uc120\ud615\uc0ac\uc0c1\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p><!-- .. --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\ubb38\uc81c 1. \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4. \\(f(x,\\,y) = \\sin x + \\sin y\\) \uc774\ub54c, \\(f\\)\uac00 \ubaa8\ub4e0 \uc810 \\((x,\\,y)\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (\\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95\uc744 \uc0ac\uc6a9\ud560 \uac83.) \ud480\uc774. \uc810 \\((x,\\,y)\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\frac{\\epsilon}{2}\\) \uc774\ub77c\uace0 \ud558\uba74 \ubaa8\ub4e0 \uc2e4\uc218 \\(s,\\) \\(t\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4. \\( \\begin{align} \\lvert s-x \\rvert < \\delta \\, &#038;\\Rightarrow \\, \\lvert&hellip;\n<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[],"class_list":["post-4482","post","type-post","status-publish","format-standard","hentry","category-calculus-ap"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4482","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=4482"}],"version-history":[{"count":56,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4482\/revisions"}],"predecessor-version":[{"id":5299,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4482\/revisions\/5299"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4482"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4482"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4482"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}