{"id":1930,"date":"2019-03-05T23:50:38","date_gmt":"2019-03-05T14:50:38","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1930"},"modified":"2019-09-05T15:03:00","modified_gmt":"2019-09-05T06:03:00","slug":"calculus-definition-of-a-limit-at-a-point","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-definition-of-a-limit-at-a-point\/","title":{"rendered":"\ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc815\uc758"},"content":{"rendered":"<p>\ud568\uc218\uc758 \uadf9\ud55c\uc740 \ubbf8\uc801\ubd84\uc758 \uc2dc\uc791\ubd80\ud130 \ub05d\uae4c\uc9c0 \ubaa8\ub4e0 \uacf3\uc5d0 \ub098\ud0c0\ub098\ub294 \uac1c\ub150\uc774\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc5c4\ubc00\ud55c \uc815\uc758\ub97c \uc0b4\ud3b4\ubcf4\uace0, \uadf9\ud55c\uacfc \uad00\ub828\ub41c \uae30\ubcf8\uc801\uc778 \uc131\uc9c8\uc744 \uc99d\uba85\ud55c\ub2e4. \uc774 \uae00\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ud568\uc218\ub294 \uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\uc774 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc778 \uac83\uc73c\ub85c \ud55c\uc815\ud55c\ub2e4.<\/p>\n<h3>\uadf9\ud55c\uc758 \uc5c4\ubc00\ud55c \uc815\uc758<\/h3>\n<p>\\(f\\)\uac00 \uc2e4\uc218 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc9c1\uad00\uc801\uc73c\ub85c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc774 \\(L\\)\uc774\ub77c\ub294 \uac83\uc740 \\(x\\)\uac00 \\(c\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc774 \ub2e4\uac00\uac08 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \\(L\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc774 \ub2e4\uac00\uac10\uc744 \uc758\ubbf8\ud55c\ub2e4. \ub17c\ub9ac\uc801 \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1.<\/span><br \/>\n\\(c\\)\uc640 \\(L\\)\uc774 \uc2e4\uc218\uc774\uace0, \ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. (\ub2e8, \\(c\\)\uc5d0\uc11c\ub294 \\(f\\)\uc758 \uac12\uc774 \uc815\uc758\ub418\uc9c0 \uc54a\uc544\ub3c4 \ub41c\ub2e4.) \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c | < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x) - L | < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74, \u2018\\(x\\)\uac00 \\(c\\)\uc5d0 \uc811\uadfc\ud560 \ub54c \\(f\\)\uc758 \uadf9\ud55c\uc740 \\(L\\)\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294\n\\[f(x) \\,\\to\\, L \\quad \\text{as} \\quad x\\,\\to\\,c \\tag{1}\\]\n\ub610\ub294\n\\[\\lim_{x\\,\\to\\,c} f(x) = L\\tag{2}\\]\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n<p>\uadf9\ud55c\uc758 \uc815\uc758\uc5d0\uc11c \\(x\\)\ub294 \ud56d\uc0c1 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc810\uc744 \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n<p>\ub2e4\uc74c \ud45c\ud604\uc740 \ubaa8\ub450 \uac19\uc740 \ub73b\uc774\ub2e4.<\/p>\n<ul>\n<li>\\(x\\)\uac00 \\(c\\)\uc5d0 \uc811\uadfc\ud560 \ub54c \\(f\\)\uc758 \uadf9\ud55c\uc740 \\(L\\)\uc774\ub2e4.<\/li>\n<li>\\(x=c\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc740 \\(L\\)\uc774\ub2e4.<\/li>\n<li>\\(x\\)\uac00 \\(c\\)\uc5d0 \uc811\uadfc\ud560 \ub54c \\(f(x)\\)\ub294 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\\(x\\,\\to\\,c\\)\uc77c \ub54c \\(f(x) \\,\\to\\, L\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ud568\uc22b\uac12\uacfc \uadf9\ud55c\uac12\uc740 \uad00\uacc4\uac00 \uc5c6\ub2e4. \uc608\ucee8\ub300 \ub2e4\uc74c \uc138 \ud568\uc218 \ubaa8\ub450 \\(3\\)\uc5d0\uc11c \uadf9\ud55c\uac12\uc740 \\(7\\)\uc774\ub2e4.<\/p>\n<p>\\[f(x) = 7 ,\\]<\/p>\n<p>\\[g(x) = \\begin{cases}<br \/>\n7 \\quad &#038; \\text{if} \\,\\, x \\ne 3 \\\\[8pt]<br \/>\n0 \\quad &#038; \\text{if} \\,\\, x = 3 ,<br \/>\n\\end{cases}\\]<\/p>\n<p>\\[h(x) = \\frac{7(x-3)}{x-3}.<br \/>\n\\]<\/p>\n<h3>\uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uae30<\/h3>\n<p>\uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \uadf9\ud55c\uac12\uc744 \uc99d\uba85\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{x\\,\\to\\,1} (3x-1) = 2\\]<br \/>\n\\(f(x) = 3x-1 ,\\) \\(c = 1,\\) \\(L = 2\\)\ub77c\uace0 \ud558\uba74 \\(f\\)\ub294 \\(c\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc5f4\ub9b0\uad6c\uac04, \uc989 \uc2e4\uc218 \uc804\uccb4 \uad6c\uac04\uc5d0\uc11c \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4. \uc774\uc81c \uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc790.<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\frac{1}{3} \\epsilon\\)\uc774\ub77c\uace0 \ud558\uba74 \\(\\delta > 0\\)\uc774\ub2e4. \uc774\uc81c \\( 0 < | x-1 | < \\delta\\)\uc784\uc744 \uac00\uc815\ud558\uba74\n\\[|f(x) - L | = | (3x-1) -2 | = |3x-3| = 3 |x-1| < 3 \\delta = \\epsilon\\]\n\uc774\ubbc0\ub85c \\(x\\)\uac00 \\(1\\)\uc5d0 \uc811\uadfc\ud560 \ub54c \\((3x-1)\\)\uc740 \\(2\\)\uc5d0 \uc811\uadfc\ud55c\ub2e4.\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc704 \ubcf4\uae30\uc5d0\uc11c \\(\\delta\\)\uc758 \uac12\uc740 \\(\\frac{\\epsilon}{3}\\)\ubcf4\ub2e4 \ub354 \uc791\uc740 \uac83\uc73c\ub85c \ub450\uc5b4\ub3c4 \ub41c\ub2e4. \\(\\epsilon\\)\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uadf9\ud55c\uc758 \uc815\uc758\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(\\delta\\)\uc758 \uac12\uc740 \ud558\ub098\ub9cc \uc788\ub294 \uac83\uc774 \uc544\ub2c8\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\\(k\\)\uac00 \uc2e4\uc218\uc77c \ub54c, \uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{x\\,\\to\\,k} x = k\\]<br \/>\n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\epsilon\\)\uc774\ub77c\uace0 \ud558\uba74 \\(\\delta > 0\\)\uc774\ub2e4. \uc774\uc81c \\( 0 < |x-k| < \\delta\\)\uc784\uc744 \uac00\uc815\ud558\uba74 \\(|x-k| < \\delta = \\epsilon\\)\uc774\ubbc0\ub85c \uc6d0\ud558\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.\n<span class=\"qee\"><\/span>\n<\/div>\n<p>\ubcf4\uae30\uc5d0\uc11c \ubcf4\ub2e4\uc2dc\ud53c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc774 \\(L\\)\uc774\ub77c\ub294 \uc99d\uba85\uc744 \u2018\uae30\uc220\u2019\ud558\ub294 \uacfc\uc815\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4.<\/li>\n<li>\ub2e4\uc74c\uc73c\ub85c \uc801\ub2f9\ud55c \\(\\delta > 0\\)\ub97c \uc124\uc815\ud55c\ub2e4.<\/li>\n<li>\\(0 < | x-c | < \\delta\\)\uc784\uc744 \uac00\uc815\ud558\uace0 \\( |f(x) - L | < \\epsilon\\)\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc778\ub2e4.<\/li>\n<\/ul>\n<p>\uc774\uc640 \uac19\uc740 \uc99d\uba85\uc758 \uae30\uc220 \uacfc\uc815\uc5d0\uc11c \\(\\delta\\)\uc758 \uac12\uc744 \ucc3e\ub294 \uacfc\uc815\uc740 \ubcf4\ud1b5 \uc798 \ub4dc\ub7ec\ub098\uc9c0 \uc54a\ub294\ub2e4. \\(\\delta\\)\ub97c \ucc3e\ub294 \uacfc\uc815\uc740 \uc99d\uba85\uc744 \uae30\uc220\ud558\ub294 \uacfc\uc815\uc774 \uc544\ub2c8\ub77c \uc99d\uba85\uc744 \uae30\uc220\ud558\uae30 \uc704\ud55c \uc0ac\uc804 \uc791\uc5c5\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{x\\,\\to\\,2} (x^2 +1 ) = 5\\]<br \/>\n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf9\ud55c\uc758 \uc815\uc758\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(\\delta\\)\ub97c \ucc3e\uc544\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ud55c \uc591\uc218 \\(\\delta\\)\uac00 \uc774\ubbf8 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \uac00\uc815\ud558\uace0 \\(|x-2| < \\delta\\)\uc784\uc744 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74\n\\[| (x^2 +1) -5 | = |x^2 -4| = |x+2|\\,|x-2| < \\delta |x+2|\\]\n\ub97c \uc5bb\ub294\ub2e4. \ub9c8\uc9c0\ub9c9 \ud56d \\(\\delta |x+2|\\)\uc5d0\uc11c \\(\\delta\\)\uc758 \uac12\uc740 \ucda9\ubd84\ud788 \uc791\uac8c(\\(0\\)\uc5d0 \uac00\uae5d\uac8c) \ub9cc\ub4e4\uc5b4\uc904 \uc218 \uc788\uc9c0\ub9cc \\(|x+2|\\)\uc758 \uac12\uc740 \uadf8\ub807\uc9c0 \uc54a\ub2e4.<\/p>\n<p>\ub300\uc2e0 \\(x\\)\uc758 \uac12\uc758 \ubc94\uc704\uac00 \uc81c\ud55c\ub418\ub3c4\ub85d \ud558\uc5ec \\(|x+2|\\)\uc758 \uac12\uc758 \ubc94\uc704\ub3c4 \uc81c\ud55c\ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \ud558\uc9c0\ub9cc \uc6b0\ub9ac\ub294 \\(x\\)\uc758 \uac12\uc758 \ubc94\uc704\ub97c \uc9c1\uc811 \uc81c\ud55c\ud558\uc9c0 \ubabb\ud55c\ub2e4. \ub2e4\ub9cc \\(|x-2|\\)\uc758 \uac12\uc758 \ubc94\uc704\ub9cc \uc81c\ud55c\ud560 \uc218 \uc788\uc744 \ubfd0\uc774\ub2e4. \ub9cc\uc57d \\(|x-2| < 1\\)\uc774\ub77c\uba74 \\(1 < x <3\\)\uc774\ubbc0\ub85c \\(|x+2| < 5\\)\ub77c\ub294 \uc81c\ud55c\ub41c \ubc94\uc704\ub97c \uc5bb\ub294\ub2e4. \uc774\uac83\uc744 \uc704\ud574\uc11c\ub294 \\(\\delta \\le 1\\)\uc774\ub77c\ub294 \uc870\uac74\uc744 \uc8fc\uba74 \ub41c\ub2e4. (\ubb3c\ub860 \\(|x-2|\\)\uc758 \uac12\uc758 \ubc94\uc704\ub97c \uc81c\ud55c\ud558\ub294 \uc218\ub294 \ub9c8\uc74c\ub300\ub85c \uc815\ud560 \uc218 \uc788\ub2e4. \uc608\ucee8\ub300 \\(|x-2| < 1\\) \ub300\uc2e0 \\(|x-2| < 7\\)\ub85c \ub450\uc5b4\ub3c4 \ub41c\ub2e4. \uc774 \uacbd\uc6b0 \\(|x+2| < 5\\) \ub300\uc2e0 \\(|x+2| < 11\\)\uc774\ub77c\ub294 \uc81c\ud55c\ub41c \ubc94\uc704\ub97c \uc5bb\ub294\ub2e4.)<\/p>\n<p>\\(\\delta \\le 1\\)\uc774\ub77c\ub294 \uac00\uc815 \ud558\uc5d0  \\(\\delta |x+2| < \\epsilon \\)\uc774 \uc131\ub9bd\ud558\ub824\uba74 \uc5b4\ub5bb\uac8c \ud574\uc57c \ud560\uae4c? \\(|x+2| < 5\\)\uc774\ubbc0\ub85c\n\\[\\delta |x+2| < 5\\delta\\]\n\uc774\uba70, \\(\\delta \\le \\frac{\\epsilon}{5}\\)\uc774\ub77c\uba74 \uc704 \uc2dd\uc758 \uac12\uc740 \\(\\epsilon\\) \uc774\ud558\uac00 \ub41c\ub2e4. \uc774\ub85c\uc368 \\(\\delta\\)\uc758 \uac12\uc740 \ub450 \ubd80\ub4f1\uc2dd \\(\\delta \\le 1,\\) \\(\\delta \\le \\frac{\\epsilon}{5}\\)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \ucda9\ubd84\ud568\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n<p>\uc5ec\uae30\uae4c\uc9c0\ub294 \\(\\delta\\)\ub97c \ucc3e\uae30 \uc704\ud55c \ud0d0\uc0c9 \uacfc\uc815\uc774\uc5c8\uc744 \ubfd0, \uc99d\uba85\uc744 \uae30\uc220\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \uc774\uc81c \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \ub9de\uac8c \uc99d\uba85\uc744 \uae30\uc220\ud574 \ubcf4\uc790. (\uc774 \ubb38\uc81c\uac00 \uc2dc\ud5d8 \ubb38\uc81c\uc600\ub2e4\uba74, \ub2f5\uc548\uc744 \uc791\uc131\ud560 \ub550 \ub2e4\uc74c \ubb38\ub2e8\uc758 \ub0b4\uc6a9\ubd80\ud130 \uc4f0\uba74 \ub41c\ub2e4.)<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\min \\left\\{ 1 ,\\, \\frac{\\epsilon}{5} \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(0 < |x-2| < \\delta \\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\( |x-2| < 1\\)\uc774\ubbc0\ub85c \\( 1 < x < 3\\) \uc989\n\\[|x+2| < 5\\]\n\ub97c \uc5bb\ub294\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uacfc \\(\\delta \\le \\epsilon \/ 5\\)\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.\n\\[ |(x^2 +1) -5| = |x+2|\\,|x-2| < \\delta |x+2| < 5 \\delta \\le \\epsilon.\\]\n\uc774\ub85c\uc368 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \uadf9\ud55c\uc744 \uc99d\uba85\ud558\uc600\ub2e4.\n<span class=\"qee\"><\/span>\n<\/div>\n<h3>\uc218\ub834\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc774\uae30<\/h3>\n<p>\uadf9\ud55c\uc758 \uc815\uc758\uc758 \ubd80\uc815\uc744 \uc774\uc6a9\ud558\uba74 \uc8fc\uc5b4\uc9c4 \uac12\uc774 \ud568\uc218\uc758 \uadf9\ud55c\uc774 \uc544\ub2d8\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \\(c\\)\uc5d0\uc11c \ud568\uc218 \\(f : D \\to \\mathbb{R}\\)\uc758 \uadf9\ud55c\uc774 \\(L\\)\uc774\ub77c\ub294 \uac83\uc744 \ub17c\ub9ac\uae30\ud638\ub85c \ub098\ud0c0\ub0b4\uba74<br \/>\n\\[\\forall \\epsilon > 0 \\,\\, \\exists \\delta > 0 \\,\\, \\forall x \\in D \\, : \\, ( 0 < | x-c | < \\delta \\,\\rightarrow\\, |f(x)-L| < \\epsilon )\\]\n\uc774\uba70, \uc774 \ubb38\uc7a5\uc758 \ubd80\uc815\uc740\n\\[\\exists \\epsilon > 0 \\,\\, \\forall \\delta > 0 \\,\\, \\exists x \\in D \\, : \\, ( 0 < | x-c | < \\delta \\,\\,\\, \\text{and} \\,\\,\\, |f(x)-L| \\ge \\epsilon )\\]\n\uc774\ub2e4. \uc774 \ubb38\uc7a5\uc758 \ub73b\uc740 \u201c\uc801\ub2f9\ud55c \\(\\epsilon > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(\\delta > 0\\)\uc5d0 \ub300\ud558\uc5ec [\\(0 < | x-c | < \\delta\\)\uc774\uc9c0\ub9cc \\(|f(x)-L| \\ge \\epsilon\\)\uc778 \uc810 \\(x \\in D\\)\uac00 \uc874\uc7ac]\ud55c\ub2e4\u201d\uc774\ub2e4.\n<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[f(x) =\\begin{cases}<br \/>\n5 \\qquad &#038; \\text{if} \\,\\, x > 3 \\\\[8pt]<br \/>\n1 \\qquad &#038; \\text{if} \\,\\, x \\le 3<br \/>\n\\end{cases}\\]<br \/>\n\\(x = 3\\)\uc77c \ub54c \ud568\uc22b\uac12\uc740 \\(f(3) = 1\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(x \\,\\to\\,3\\)\uc77c \ub54c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc740 \\(1\\)\uc774 \uc544\ub2c8\ub2e4. \uc774\uac83\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n<p>\\(\\epsilon = 2\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\delta > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc2e4\uc218\uc758 \uc870\ubc00\uc131\uc5d0 \uc758\ud558\uc5ec \\(3 < x < 3+\\delta\\)\uc778 \uc2e4\uc218 \\(x\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774 \ub54c \\(0 < |x-3| < \\delta\\)\uc774\uc9c0\ub9cc \\[|f(x)-1| = |5-1| = 4 \\ge 2 = \\epsilon\\]\uc774\ubbc0\ub85c \\(x \\,\\to\\,3\\)\uc77c \ub54c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc740 \\(1\\)\uc774 \uc544\ub2c8\ub2e4.\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc704 \ubcf4\uae30\uc5d0\uc11c \ubcf4\ub294 \ubc14\uc640 \uac19\uc774 \uadf9\ud55c\uc758 \ubd80\uc815\uc744 \uc774\uc6a9\ud558\uc5ec \uc8fc\uc5b4\uc9c4 \uac12\uc774 \uadf9\ud55c\uac12\uc774 \uc544\ub2d8\uc744 \ubcf4\uc77c \ub54c\uc5d0\ub294 \u2018\uc801\ub2f9\ud55c \\(\\epsilon\\)\u2019\uc744 \uc7a1\ub294 \uac83\uacfc \u2018\uc801\ub2f9\ud55c \\(x\\)\u2019\ub97c \ucc3e\ub294 \uac83\uc774 \uc99d\uba85\uc758 \ud575\uc2ec\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.<\/span><br \/>\n\ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[f(x) =  \\begin{cases}<br \/>\n1 \\qquad &#038; \\text{if} \\,\\, x\\in\\mathbb{Q} \\\\[8pt]<br \/>\n0 \\qquad &#038; \\text{if} \\,\\, x\\notin\\mathbb{Q}<br \/>\n\\end{cases}\\]<br \/>\n\uadf8\ub9ac\uace0 \uc2e4\uc218 \\(c\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(x\\,\\to\\,c\\)\uc77c \ub54c \\(f(x)\\)\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc74c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n<p>\uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(x\\,\\to\\,c\\)\uc77c \ub54c \\(f(x)\\)\uc758 \uadf9\ud55c\uac12\uc774 \uc874\uc7ac\ud558\uace0, \uadf8 \uac12\uc774 \\(L\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(\\epsilon = 1\/2\\)\uc774\ub77c\uace0 \ud558\uace0, \\(\\delta > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<\/p>\n<p>\ub9cc\uc57d \\(L \\ge 1\/2\\)\uc774\ub77c\uba74 \uc5f4\ub9b0\uad6c\uac04 \\((c-\\delta ,\\, c+\\delta )\\)\uc5d0\uc11c \ubb34\ub9ac\uc218 \\(x\\)\ub97c \ud0dd\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(0 < |x-c| < \\delta\\)\uc774\uc9c0\ub9cc \\(|f(x) - L| = L \\ge \\epsilon\\)\uc774\ubbc0\ub85c \\(L\\)\uc740 \uadf9\ud55c\uac12\uc774 \uc544\ub2c8\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(L < 1\/2\\)\uc774\ub77c\uba74 \uc5f4\ub9b0\uad6c\uac04 \\((c-\\delta ,\\, c+\\delta )\\)\uc5d0\uc11c \uc720\ub9ac\uc218 \\(x\\)\ub97c \ud0dd\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(0 < |x-c| < \\delta\\)\uc774\uc9c0\ub9cc \\(|f(x) - L| = 1-L \\ge \\epsilon\\)\uc774\ubbc0\ub85c \\(L\\)\uc740 \uadf9\ud55c\uac12\uc774 \uc544\ub2c8\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \\(x\\,\\to\\,c\\)\uc77c \ub54c \uc5b4\ub5a0\ud55c \uac12\ub3c4 \\(f(x)\\)\uc758 \uadf9\ud55c\uac12\uc774 \ub420 \uc218 \uc5c6\ub2e4.<span class=\"qee\"><\/span><\/p>\n<\/div>\n<h3>\uadf9\ud55c\uc758 \uc720\uc77c\uc131<\/h3>\n<p>\uc0ac\uc2e4 (2)\uc640 \uac19\uc774 \ub4f1\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \uadf9\ud55c \uae30\ud638\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \uc774\uc720\ub294 \uadf9\ud55c\uc758 \uc720\uc77c\uc131\uc774 \ubcf4\uc7a5\ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \uc989 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span> \ud55c \uc810\uc5d0\uc11c \ud568\uc218\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4\uba74, \uadf8\uac83\uc740 \uc720\uc77c\ud558\ub2e4. \uc989<br \/>\n\\[f(x)\\,\\to\\,L \\quad \\text{as} \\quad x\\,\\to\\,c\\]<br \/>\n\uc774\uace0<br \/>\n\\[f(x)\\,\\to\\,M \\quad \\text{as} \\quad x\\,\\to\\,c\\]<br \/>\n\uc774\uba74 \\(L=M\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\frac{\\epsilon}{2}\\)\ub3c4 \uc591\uc218\uc774\ub2e4. \uc774 \ub54c \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\( 0 < |x-c| < \\delta_1 \\)\uc77c \ub54c\ub9c8\ub2e4\n\\[|f(x) - L | < \\frac{\\epsilon}{2}\\]\n\uc774 \uc131\ub9bd\ud558\uba70, \\(\\delta_2 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\( 0 < |x-c| < \\delta_2 \\)\uc77c \ub54c\ub9c8\ub2e4\n\\[|f(x) - M | < \\frac{\\epsilon}{2}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\( 0 < |x-c| < \\min \\left\\{ \\delta_1 ,\\, \\delta_2 \\right\\}\\)\uc778 \\(x\\)\ub97c \ud0dd\ud558\uba74\n\\[\\begin{align}\n|L-M| &#038;= |(L-f(x)) + (f(x)-M)| \\\\[8pt]\n&#038;\\le |f(x) - L | + |f(x)-M|  \\\\[6pt]\n&#038;< \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\n\\end{align}\\]\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\epsilon\\)\uc740 \uc784\uc758\uc758 \uc591\uc218\uc774\ubbc0\ub85c \\(|L - M| =0\\) \uc989 \\(L = M\\)\uc744 \uc5bb\ub294\ub2e4.\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<h3>\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8<\/h3>\n<p>\uadf9\ud55c\uc758 \uc815\uc758\ub294 \uadf9\ud55c\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc9c1\uc811\uc801\uc73c\ub85c \uc81c\uacf5\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc774\ubbf8 \uadf9\ud55c\uac12\uc744 \uc54c\uace0 \uc788\ub294 \ud568\uc218\ub97c \ub300\uc218\uc801\uc73c\ub85c \ubcc0\ud615\ud558\uac70\ub098 \uacb0\ud569\ud558\uc5ec \uc0c8\ub85c\uc6b4 \ud568\uc218\ub97c \ub9cc\ub4e4\uc5c8\uc744 \ub54c \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uadf9\ud55c\uac12\uc744 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8)<\/span><br \/>\n\ub9cc\uc57d \\(L,\\) \\(M,\\) \\(c,\\) \\(k\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(f\\)\uc640 \\(g\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc758 \\(c\\)\ub97c \uc81c\uc678\ud55c \uc810\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\uc73c\uba70,<br \/>\n\\[\\lim_{x\\,\\to\\,c} f(x) = L , \\quad \\lim_{x\\,\\to\\,c}g(x)=M\\]<br \/>\n\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>(a) \\((f+g)\\)\ub294 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0<br \/>\n\\[<br \/>\n\\lim _ { x\\,\\to\\,c} (f(x)+g(x)) = L+M.<br \/>\n\\]\n<\/p>\n<p>(b) \\((f-g)\\)\ub294 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0<br \/>\n\\[<br \/>\n\\lim _ { x\\,\\to\\,c} (f(x)-g(x)) = L-M .<br \/>\n\\]\n<\/p>\n<p>(c) \\(kf \\)\ub294 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0<br \/>\n\\[<br \/>\n\\lim _ { x\\,\\to\\,c} (k f(x)) = kL .<br \/>\n\\]\n<\/p>\n<p>(d) \\(fg\\)\ub294 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0<br \/>\n\\[<br \/>\n\\lim _ { x\\,\\to\\,c} (f(x)g(x)) = LM .<br \/>\n\\]\n<\/p>\n<p>(e) \\(M \\ne 0\\)\uc774\uba74 \\(f\/g\\)\ub294 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0<br \/>\n\\[\\lim_{x\\,\\to\\,c} \\frac{f(x)}{g(x)} = \\frac{L}{M} .\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>(a)\uc640 (d)\ub97c \uba3c\uc800 \uc99d\uba85\ud558\uace0, \uc774\uac83\uc744 \uc774\uc6a9\ud558\uc5ec (c)\ub97c \uc99d\uba85\ud558\uc5ec, \uadf8 \ub2e4\uc74c\uc5d0 (b)\ub97c \uc99d\uba85\ud55c\ub2e4. (e)\ub294 \uac00\uc7a5 \ub9c8\uc9c0\ub9c9\uc5d0 \uc99d\uba85\ud55c\ub2e4.<\/p>\n<p><span class=\"proof\">(a)\uc758 \uc99d\uba85.<\/span> \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_1\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\[|f(x)-L| < \\frac{\\epsilon}{2}\\]\uc774 \uc131\ub9bd\ud558\uba70, \\(\\delta_2 > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_2\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\[|g(x)-M| < \\frac{\\epsilon}{2}\\]\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\delta = \\min\\left\\{ \\delta_1 ,\\, \\delta_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\( 0 < |x-c| < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\begin{align}\n|(f(x)+g(x)) - (L+M)| &#038;= |(f(x)-L) + (g(x)-M)| \\\\[6pt]\n&#038;\\le |f(x)-L| + |g(x)-M| \\\\[6pt]\n&#038;< \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon .\n\\end{align}\\]\n<\/p>\n<p><span class=\"proof\">(d)\uc758 \uc99d\uba85.<\/span> \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(1\\)\uc740 \uc591\uc218\uc774\ubbc0\ub85c, \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_1\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ | g(x)-M | < 1 \\tag{3}\\]\n\uc774 \uc131\ub9bd\ud558\uba70, \\(\\delta_2 > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_2\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ | f(x)-L | < 1 \\tag{4}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c (3)\uacfc (4)\ub97c \ubcc0\ud615\ud558\uba74 \uac01\uac01\n\\[ |g(x)| < |M|+1 ,\\quad |f(x)| < |L|+1 \\tag{5}\\]\n\uc744 \uc5bb\ub294\ub2e4. \ud55c\ud3b8 \\( \\epsilon\/(2(|M|+1))\\)\uacfc \\(\\epsilon\/(2(|L|+1))\\)\uc740 \ubaa8\ub450 \uc591\uc218\uc774\ubbc0\ub85c, \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_3 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_3\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ |f(x) - L| < \\frac{\\epsilon}{2(|M| +1)} \\tag{6}\\]\n\uc774 \uc131\ub9bd\ud558\uba70, \\(\\delta_4 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_4\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ |g(x) - M| < \\frac{\\epsilon}{2(|L| +1)} \\tag{7}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\uc81c \\(\\delta = \\min\\left\\{ \\delta_1 ,\\, \\delta_2 ,\\, \\delta_3 ,\\, \\delta_4 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 (5), (6), (7)\uc5d0 \uc758\ud558\uc5ec\n\\[\\begin{align}\n | f(x)g(x) - LM| &#038;= | f(x)g(x) - Lg(x) + Lg(x) - LM| \\\\[8pt]\n&#038;\\le |f(x)-L| \\, |g(x)| + |L| \\, |g(x)-M| \\\\[8pt]\n&#038;< |f(x) -L| (|M|+1 ) + (|L|+1)|g(x)-M| \\\\[8pt]\n&#038;< \\frac{\\epsilon}{2(|M|+1)} (|M|+1) + (|L|+1) \\frac{\\epsilon}{2(|L|+1)} \\\\[6pt]\n&#038;= \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\n\\end{align}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<p><span class=\"proof\">(c)\uc758 \uc99d\uba85.<\/span> \\(g(x) = k\\)\ub77c\uace0 \ud558\uba74 \\(x\\,\\to\\,c\\)\uc77c \ub54c \\(g(x) \\,\\to\\,k\\)\uc774\uace0, \\(kf(x) = g(x)f(x)\\)\uc774\ubbc0\ub85c (d)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\lim _ { x\\,\\to\\,c} (kf(x)) &#038;= \\lim _ { x\\,\\to\\,c} (g(x)f(x)) \\\\[6pt]<br \/>\n&#038;= \\left( \\lim _ { x\\,\\to\\,c} g(x) \\right) \\left( \\lim _ { x\\,\\to\\,c} f(x) \\right) \\\\[6pt]<br \/>\n&#038;= kL<br \/>\n\\end{align}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p><span class=\"proof\">(b)\uc758 \uc99d\uba85.<\/span> (a)\uc640 (c)\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\lim _ { x\\,\\to\\,c} (f(x)-g(x)) &#038;= \\lim _ { x\\,\\to\\,c} (f(x) + (-1)g(x)) \\\\[6pt]<br \/>\n&#038;=\\lim _ { x\\,\\to\\,c} f(x) + \\lim _ { x\\,\\to\\,c} ((-1)g(x)) \\\\[6pt]<br \/>\n&#038;=\\lim _ { x\\,\\to\\,c} f(x) + (-1) \\lim _ { x\\,\\to\\,c} g(x) \\\\[6pt]<br \/>\n&#038;= L + (-1)M \\\\[8pt]<br \/>\n&#038;= L-M.<br \/>\n\\end{align}\\]\n<\/p>\n<p><span class=\"proof\">(e)\uc758 \uc99d\uba85.<\/span><br \/>\n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(|M|\/2\\)\uc740 \uc591\uc218\uc774\ubbc0\ub85c \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_1\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[|g(x) - M| < \\frac{|M|}{2}\\tag{8}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \\(|M|^2 \\epsilon \/ 2\\)\uc740 \uc591\uc218\uc774\ubbc0\ub85c \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_2 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_2\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[|g(x) - M| < \\frac{|M|^2}{2} \\epsilon\\tag{9}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\delta = \\min\\left\\{ \\delta_1 ,\\,\\delta_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(0 < |x-c| < \\delta\\)\ub77c\uace0 \ud558\uc790. (8)\uc744 \ubcc0\ud615\ud558\uba74\n\\[|g(x)| > \\frac{|M|}{2}\\]<br \/>\n\uc989<br \/>\n\\[\\frac{1}{|g(x)|} < \\frac{2}{|M|}\\tag{10}\\]\n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (10), (9)\uc5d0 \uc758\ud558\uc5ec\n\\[\\begin{align}\n\\left| \\frac{1}{g(x)} - \\frac{1}{M} \\right| &#038;= \\frac{|M - g(x)|}{|g(x)|\\,|M|} \\\\[6pt]\n&#038;< \\frac{2}{|M|^2} |M-g(x)| \\\\[6pt]\n&#038;< \\frac{2}{|M|^2} \\cdot \\frac{|M|^2}{2} \\epsilon = \\epsilon\n\\end{align}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989\n\\[\\lim_{x\\,\\to\\,c}\\frac{1}{g(x)} = \\frac{1}{M}\\]\n\uc774\ub2e4. \uc774 \ub4f1\uc2dd\uacfc (b)\ub97c \uc774\uc6a9\ud558\uba74\n\\[\\lim_{x\\,\\to\\,c}\\frac{f(x)}{g(x)} = \\lim_{x\\,\\to\\,c}\\left( f(x) \\cdot \\frac{1}{g(x)}\\right) = L \\cdot \\frac{1}{M} = \\frac{L}{M}\\]\n\uc744 \uc5bb\ub294\ub2e4.\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<p><!--\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8)<\/span>\n\ub9cc\uc57d \\(L,\\) \\(c\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uba70\n\\[\\lim_{x\\,\\to\\,c} f(x) = L \\]\n\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n\n\n\n\n<p>(a) \\(f^n\\)\uc740 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0\n\\[\\lim _ { x\\,\\to\\,c} [f(x)]^n = L^n .\\]\n<\/p>\n\n\n\n\n\n<p>(b) \\(n\\)\uc774 \uc9dd\uc218\uc774\uace0 \\(L > 0\\)\uc774\uba74 \\(\\sqrt[n]{f}\\)\ub294 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0\n\\[\\lim _ { x\\,\\to\\,c} \\sqrt[n]{f(x)} = \\sqrt[n]{L} .\\]\n<\/p>\n\n\n\n\n\n<p>(c) \\(n\\)\uc774 \ud640\uc218\uc774\uba74 \\(\\sqrt[n]{f}\\)\ub294 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uace0\n\\[\\lim _ { x\\,\\to\\,c} \\sqrt[n]{f(x)} = \\sqrt[n]{L} .\\]\n<\/p>\n\n\n\n<\/div>\n\n\n--><\/p>\n<p>\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\ud56d\ud568\uc218\uc640 \ubd84\uc218\ud568\uc218\uc758 \uadf9\ud55c\uac12\uc744 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.<\/span><br \/>\n\ub2e4\ud56d\ud568\uc218\uc758 \uadf9\ud55c\uac12\uc740 \ud568\uc22b\uac12\uacfc \uac19\ub2e4:<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{x\\to3}(2x^3 -x^2 +4x )<br \/>\n&#038;= \\lim_{x\\to 3} (2x^3) &#8211; \\lim_{x\\to 3}(x^2 ) + \\lim_{x\\to 3}(4x) \\\\[6pt]<br \/>\n&#038;= 2 \\lim_{x\\to 3} x^3 &#8211; \\lim_{x\\to 3}(x^2 ) + 4 \\lim_{x\\to 3} x \\\\[4pt]<br \/>\n&#038;= 2 \\left(\\lim_{x\\to 3} x\\right)^3 &#8211; \\left(\\lim_{x\\to 3}x\\right)^2 +4 \\lim_{x\\to 3} x \\\\[6pt]<br \/>\n&#038;= 2 \\times 3^3 &#8211; 3^2 + 4 \\times 3 \\\\[8pt]<br \/>\n&#038;= 54 &#8211; 9 + 12 = 57.<br \/>\n\\end{align}\\]\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 7.<\/span><br \/>\n\ubd84\uc218\ud568\uc218\uc758 \uadf9\ud55c\uac12\uc740, \ubd84\ubaa8\uac00 \\(0\\)\uc774 \uc544\ub2c8\ub77c\uba74, \ud568\uc22b\uac12\uacfc \uac19\ub2e4:<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{x\\to -1}\\frac{x^2 -3}{x+4}<br \/>\n&#038;= \\frac{\\lim_{x\\to -1} (x^2 -3)}{\\lim_{x\\to -1} (x+4)} \\\\[6pt]<br \/>\n&#038;= \\frac{(-1)^2 -3}{(-1)+4} \\\\[6pt]<br \/>\n&#038;= \\frac{-2}{3} = &#8211; \\frac{2}{3} .<br \/>\n\\end{align}\\]\n<\/div>\n<h3>\ubd80\ub4f1\ud638\uc640 \uad00\ub828\ub41c \uadf9\ud55c\uc758 \uc131\uc9c8<\/h3>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \ud568\uc218\uc758 \uc2dd\uc744 \uc815\ud655\ud558\uac8c \uc54c\uc9c0 \ubabb\ud558\uac70\ub098, \ud568\uc218\uc758 \uc2dd\uc744 \ub300\uc218\uc801\uc73c\ub85c \ud480\uae30 \uc5b4\ub824\uc6b8 \ub54c\uc5d0\ub3c4 \uadf9\ud55c\uac12\uc744 \uad6c\ud560 \uc218 \uc788\uac8c \ud574\uc8fc\ub294 \uc815\ub9ac\uc774\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac)<\/span><br \/>\n\\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0\uad6c\uac04\uc758 \\(x=c\\)\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[g(x) \\le f(x) \\le h(x)\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uace0<br \/>\n\\[\\lim_{x\\to c} g(x) = \\lim_{x\\to c} h(x) = L\\]<br \/>\n\uc774\uba74<br \/>\n\\[\\lim_{x\\to c} f(x) = L\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_1\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[|g(x) - L| < \\epsilon\\]\n\uc989\n\\[ L - \\epsilon < g(x) < L+ \\epsilon\\]\n\uc774 \uc131\ub9bd\ud558\uba70, \\(\\delta_2 > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_2\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[|h(x) - L| < \\epsilon\\]\n\uc989\n\\[ L - \\epsilon < h(x) < L+ \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\delta = \\min\\left\\{\\delta_1 ,\\,\\delta_2 \\right\\}\\)\ub77c\uace0 \ud558\uba74 \\(0 < |x-c| < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ L - \\epsilon < g(x) \\le f(x) \\le h(x) < L + \\epsilon\\]\n\uc989\n\\[ | f(x) - L | < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c \\(x\\,\\to\\,c\\)\uc77c \ub54c \\(f(x)\\,\\to\\,L\\)\uc774\ub2e4.<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<p>\uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \ubcf4\uae30\uc640 \uac19\uc774 \uc9c1\uc811 \uad6c\ud558\uae30 \uc5b4\ub824\uc6b4 \ud568\uc218\uc758 \uadf9\ud55c\uac12\uc744 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 8.<\/span><br \/>\n\\(\\theta\\)\uac00 \\(0\\)\uc774 \uc544\ub2cc \uac01\uc774\uace0 \uadf8 \ub2e8\uc704\uac00 \ub77c\ub514\uc548\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/03\/\uadf9\ud55c\uc758-\uc815\uc758-\uc0ac\uc778\ucf54\uc0ac\uc778\ubd80\ub4f1\uc2dd-300x295.png\" alt=\"\" width=\"300\" height=\"295\" class=\"aligncenter size-medium wp-image-2287\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/03\/\uadf9\ud55c\uc758-\uc815\uc758-\uc0ac\uc778\ucf54\uc0ac\uc778\ubd80\ub4f1\uc2dd-300x295.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/03\/\uadf9\ud55c\uc758-\uc815\uc758-\uc0ac\uc778\ucf54\uc0ac\uc778\ubd80\ub4f1\uc2dd.png 492w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>\uadf8\ub7ec\uba74 \uc704 \uadf8\ub9bc\uc744 \ud1b5\ud574 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[- | \\theta | \\le \\sin \\theta \\le | \\theta | ,\\\\[8pt]<br \/>\n&#8211; | \\theta | \\le 1- \\cos \\theta \\le | \\theta | .\\]<br \/>\n\uc774 \ubd80\ub4f1\uc2dd\uacfc \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \uadf9\ud55c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\lim_{\\theta \\,\\to\\,0} \\sin \\theta =0 ,\\\\[8pt]<br \/>\n\\lim_{\\theta \\,\\to\\,0} \\cos\\theta = 1.\\]\n<\/p>\n<\/div>\n<p>\ub450 \ud568\uc218\uc758 \ub300\uc18c \uad00\uacc4\uc640 \uadf9\ud55c\uac12\uc758 \ub300\uc18c \uad00\uacc4 \uc0ac\uc774\uc5d0\ub294 \ub2e4\uc74c \uc815\ub9ac\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.<\/span><br \/>\n\\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0\uad6c\uac04\uc758 \\(x=c\\)\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[f(x) \\le g(x)\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uace0, \\(x\\,\\to\\,c\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(g(x)\\)\uc758 \uadf9\ud55c\uc774 \ubaa8\ub450 \uc874\uc7ac\ud558\uba74<br \/>\n\\[\\lim_{x\\to c} f(x) \\le \\lim_{x\\to c}g(x)\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uacb0\ub860\uc744 \ubd80\uc815\ud558\uc790. \uc989<br \/>\n\\[\\lim_{x\\to c}f(x) = L ,\\quad \\lim_{x\\to c}g(x)=M\\]<br \/>\n\uc774\uace0 \\(L > M\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790.<br \/>\n\\[\\epsilon = \\frac{L-M}{2}\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_1\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[|f(x)-L| < \\epsilon = \\frac{L-M}{2}\\]\n\uc989\n\\[f(x) > L &#8211; \\frac{L-M}{2}\\]<br \/>\n\uc774 \uc131\ub9bd\ud558\uba70, \\(\\delta_2 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta_2\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[|g(x)-M| < \\epsilon = \\frac{L-M}{2}\\]\n\uc989\n\\[g(x) < M + \\frac{L-M}{2}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\delta = \\min\\left\\{\\delta_1 ,\\,\\delta_2\\right\\}\\)\ub77c\uace0 \ud558\uba74 \\(0 < |x-c| < \\delta\\)\uc778 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[f(x) > L &#8211; \\frac{L-M}{2} = \\frac{L+M}{2} = M + \\frac{L-M}{2} > g(x)\\]<br \/>\n\uc774\ubbc0\ub85c, \\(f(x) \\le g(x)\\)\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(L \\le M\\)\uc774\ub2e4.<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<h3>\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uad00\uacc4<\/h3>\n<p>\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \ubc00\uc811\ud55c \uad00\uacc4\ub97c \uac00\uc9c0\uace0 \uc788\ub2e4. \ub450 \uadf9\ud55c\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uae30 \uc704\ud558\uc5ec \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2. (\uc218\uc5f4\uc758 \uadf9\ud55c)<\/span><\/p>\n<p>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc218\uc5f4\uc774\uace0 \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc778 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert x_n &#8211; L \\rvert < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74, \u2018\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 \\(L\\)\uc5d0 <span class=\"defined\">\uc218\ub834<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0 \\(L\\)\uc744 \\(\\left\\{x_n \\right\\}\\)\uc758 <span class=\"defined\">\uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc5b4\ub5a0\ud55c \uac12\uc5d0\ub3c4 \uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c \u2018\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\ubc1c\uc0b0<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\lim_{n\\to\\infty} x_n = L\\]<br \/>\n\ub610\ub294<br \/>\n\\[x_n \\,\\to\\, L .\\]\n<\/p>\n<p>\uc218\uc5f4\uc758 \uadf9\ud55c \ub610\ud55c \ud568\uc218\uc758 \uadf9\ud55c\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc815\ub9ac 2\uc640 \uc815\ub9ac 3\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ub300\uc218\uc801 \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\ub2e4. \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uc744 \uc0c1\uc138\ud788 \uc0b4\ud3b4\ubcf4\ub294 \uac83\uc740 \uc774 \uae00\uc758 \ubaa9\uc801\uc774 \uc544\ub2c8\ubbc0\ub85c \uc774 \uae00\uc5d0\uc11c \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uacfc \uad00\ub828\ub41c \ub17c\uc758\ub294 \uc0dd\ub7b5\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 5. (\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uad00\uacc4)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c, \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \\(L\\)\uc5d0 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubaa8\ub4e0 \ud56d\uc774 \\(I \\setminus \\left\\{ c \\right\\}\\)\uc5d0 \uc18d\ud558\uace0 \\(x_n \\,\\to\\, c\\)\uc778 \uc784\uc758\uc758 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\( f( x_n ) \\,\\to\\, L\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uba3c\uc800 \ud544\uc694\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(I\\setminus \\left\\{ c \\right\\}\\)\uc5d0 \uc18d\ud558\uace0 \\(c\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\uc81c \\(f(x_n ) \\,\\to\\, L\\)\uc784\uc744 \uc99d\uba85\ud574\uc57c \ud55c\ub2e4.<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(c\\)\uc5d0\uc11c \\(f\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\( 0 < \\lvert x-c \\rvert < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert f(x)-L \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc778 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert x_n &#8211; c \\rvert < \\delta\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub3d9\uc77c\ud55c \\(N\\)\uc5d0 \ub300\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \\(\\lvert f(x_n ) &#8211; L \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c \\(f(x_n ) \\,\\to\\,L\\)\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \ucda9\ubd84\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \\(L\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\epsilon > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \uc784\uc758\uc758 \\(\\delta > 0\\)\uc5d0 \ub300\ud558\uc5ec \\( 0 < \\lvert x-c \\rvert < \\delta\\)\uc784\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \\(\\lvert f(x) - L \\rvert \\ge \\epsilon\\)\uc778 \\(x\\)\uac00 \\(I\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \\(\\delta_n = 1\/n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta_n > 0\\)\uc774\ubbc0\ub85c, \\(0 < \\lvert x_n - c \\rvert < \\delta_n\\)\uc784\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \\(\\lvert f(x_n ) - L \\rvert \\ge \\epsilon\\)\uc778 \uc810 \\(x_n\\)\uc774 \\(I\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 \\(c\\)\uc5d0 \uc218\ub834\ud558\uc9c0\ub9cc \\(\\left\\{ f(x_n ) \\right\\}\\)\uc740 \\(L\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 9.<\/span><br \/>\n\ubcf4\uae30 5\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud574 \ubcf4\uc790. \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[f(x) =  \\begin{cases}<br \/>\n1 \\qquad &#038; \\text{if} \\,\\, x\\in\\mathbb{Q} \\\\[8pt]<br \/>\n0 \\qquad &#038; \\text{if} \\,\\, x\\notin\\mathbb{Q}<br \/>\n\\end{cases}\\]<br \/>\n\uadf8\ub9ac\uace0 \uc2e4\uc218 \\(c\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(x\\,\\to\\,c\\)\uc77c \ub54c \\(f(x)\\)\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc74c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n<p>\uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(x\\,\\to\\,c\\)\uc77c \ub54c \\(f(x)\\)\uc758 \uadf9\ud55c\uac12\uc774 \uc874\uc7ac\ud558\uace0, \uadf8 \uac12\uc774 \\(L\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790.<\/p>\n<p>\uc720\ub9ac\uc218\uc758 \uc870\ubc00\uc131\uc5d0 \uc758\ud558\uc5ec \\(c\\)\uc5d0 \uc218\ub834\ud558\uace0 \\(q_n \\ne c\\)\uc778 \uc720\ub9ac\uc218\uc5f4 \\(\\left\\{ q_n \\right\\}\\)\uc774 \uc874\uc7ac\ud558\uba70, \ubb34\ub9ac\uc218\uc758 \uc870\ubc00\uc131\uc5d0 \uc758\ud558\uc5ec \\(c\\)\uc5d0 \uc218\ub834\ud558\uace0 \\(r_n \\ne c\\)\uc778 \ubb34\ub9ac\uc218\uc5f4 \\(\\left\\{ r_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \uc815\ub9ac 5\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[L = \\lim_{x\\to c} f(x) = \\lim_{n\\to\\infty} f(q_n) = \\lim_{n\\to\\infty} 1 = 1\\]<br \/>\n\uadf8\ub9ac\uace0<br \/>\n\\[L = \\lim_{x\\to c} f(x) = \\lim_{n\\to\\infty} f(r_n) = \\lim_{n\\to\\infty} 0 = 0\\]<br \/>\n\uc774\ubbc0\ub85c \\(1 = L = 0\\)\uc774\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(c\\)\uc5d0\uc11c \\(f(x)\\)\uc758 \uadf9\ud55c\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<h3>Meme<\/h3>\n<p><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/03\/math-meme-epsilon-delta.jpg\" alt=\"Meme: Epsilon-Delta\" width=\"512\" height=\"1152\" class=\"aligncenter size-full wp-image-3047\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/03\/math-meme-epsilon-delta.jpg 512w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/03\/math-meme-epsilon-delta-133x300.jpg 133w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/03\/math-meme-epsilon-delta-455x1024.jpg 455w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/><\/p>\n<p class=\"aligncenter\">(Source: <a href=\"https:\/\/twitter.com\/cworley65\/status\/1144724423175561221?s=20\">Chase Worley&#8217;s Twitter<\/a>)<\/p>\n<p>\uc0ac\uc2e4 \uc704 \ubc08\uc758 \ub0b4\uc6a9\uc740 \uc798\ubabb\ub418\uc5c8\ub2e4. \uc989 \uc138 \ubc88\uc9f8 \ucef7\uc5d0\uc11c \u2018\\(\\delta\\)\u2019\ub97c \u2018\\(\\delta > 0\\)\u2019\uc73c\ub85c \ubc14\uafb8\uc5b4\uc57c \ub9de\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(\\delta = -1\\)\uc774\ub77c\uace0 \ub450\uba74 \u2018\uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(0 < \\lvert x-c \\rvert < \\delta\\)\ub294 \uac70\uc9d3\u2019\uc774\ubbc0\ub85c \\(f,\\) \\(c,\\) \\(L\\)\uc5d0 \uc0c1\uad00 \uc5c6\uc774\n\\[ ( 0 < \\lvert x-c \\rvert < \\delta) \\,\\,\\,\\to\\,\\,\\, (\\lvert f(x) - L \\rvert < \\epsilon) \\]\n\uc774 \uacf5\uc870\uac74\uc801\uc73c\ub85c(\uac00\uc815\uc774 \uac70\uc9d3\uc774\ubbc0\ub85c \uacb0\ub860\uc5d0 \uc0c1\uad00 \uc5c6\uc774) \ucc38\uc774 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p><!-- -- --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\ud568\uc218\uc758 \uadf9\ud55c\uc740 \ubbf8\uc801\ubd84\uc758 \uc2dc\uc791\ubd80\ud130 \ub05d\uae4c\uc9c0 \ubaa8\ub4e0 \uacf3\uc5d0 \ub098\ud0c0\ub098\ub294 \uac1c\ub150\uc774\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc5c4\ubc00\ud55c \uc815\uc758\ub97c \uc0b4\ud3b4\ubcf4\uace0, \uadf9\ud55c\uacfc \uad00\ub828\ub41c \uae30\ubcf8\uc801\uc778 \uc131\uc9c8\uc744 \uc99d\uba85\ud55c\ub2e4. \uc774 \uae00\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ud568\uc218\ub294 \uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\uc774 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc778 \uac83\uc73c\ub85c \ud55c\uc815\ud55c\ub2e4. \uadf9\ud55c\uc758 \uc5c4\ubc00\ud55c \uc815\uc758 \\(f\\)\uac00 \uc2e4\uc218 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc9c1\uad00\uc801\uc73c\ub85c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc774 \\(L\\)\uc774\ub77c\ub294 \uac83\uc740 \\(x\\)\uac00 \\(c\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc774 \ub2e4\uac00\uac08 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \\(L\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc774 \ub2e4\uac00\uac10\uc744 \uc758\ubbf8\ud55c\ub2e4.&hellip;<\/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":[140,141,127,142,131,144,126,138,130,137,143,139],"class_list":["post-1930","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-convergence","tag-divergence","tag-limit","tag-oscillation","tag-sandwich-theorem","tag-uniqueness","tag-126","tag-138","tag-130","tag-137","tag-143","tag-139"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1930","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=1930"}],"version-history":[{"count":157,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1930\/revisions"}],"predecessor-version":[{"id":3230,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1930\/revisions\/3230"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1930"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1930"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1930"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}