{"id":1932,"date":"2019-03-05T23:53:01","date_gmt":"2019-03-05T14:53:01","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1932"},"modified":"2019-09-05T15:03:58","modified_gmt":"2019-09-05T06:03:58","slug":"calculus-one-sided-limits","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-one-sided-limits\/","title":{"rendered":"\uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c"},"content":{"rendered":"

\\(x\\)\ucd95\uc5d0\uc11c \ud55c \uc810 \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \uc218 \uc788\ub294 \ubc29\ud5a5\uc740 \ub450 \uac00\uc9c0\uac00 \uc788\ub2e4. \uc989 \\(c\\)\uc758 \uc67c\ucabd\uc5d0\uc11c \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \uc218\ub3c4 \uc788\uc73c\uba70, \\(c\\)\uc758 \uc624\ub978\ucabd\uc5d0\uc11c \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \uc218\ub3c4 \uc788\ub2e4. \uc774\uc640 \uac19\uc774 \\(x\\)\uac00 \\(c\\)\uc5d0 \ub2e4\uac00\uac00\ub294 \ubc29\ud5a5\uc5d0 \ub530\ub77c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uc744 \uad6c\ubd84\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc758 \uc815\uc758<\/h3>\n

\\(x\\)\uac00 \\(c\\)\uc758 \uc67c\ucabd\uc5d0\uc11c \\(c\\)\uc5d0 \uc811\uadfc\ud560 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \\(L\\)\uc5d0 \ub2e4\uac00\uac00\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc88c\uadf9\ud55c \\(L\\)\uc744 \uac00\uc9c4\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \uc6b0\uadf9\ud55c\uc5d0 \ub300\ud574\uc11c\ub3c4 \ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \ub17c\ub9ac\uc801\uc778 \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\uc758 1. (\uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c)<\/span><\/p>\n

    \n
  1. \\(L\\)\uacfc \\(c\\)\uac00 \uc2e4\uc218\uc774\uace0 \ud568\uc218 \\(f : D \\to \\mathbb{R}\\)\uac00 \\(c\\)\ub97c \uc624\ub978\ucabd \ub05d\uc810\uc73c\ub85c \ud558\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < c-x < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in D\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x)-L| < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc88c\uadf9\ud55c \\(L\\)\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0,\n\\[f(x) \\,\\to\\, L \\quad \\text{as} \\quad x \\,\\to\\, c^{-}\\]\n\ub610\ub294\n\\[\\lim_{x\\,\\to\\,c^{-}} f(x) = L\\]\n\ub610\ub294\n\\[f(c^- ) = L\\]\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/li>\n
  2. \\(L\\)\uacfc \\(c\\)\uac00 \uc2e4\uc218\uc774\uace0 \ud568\uc218 \\(f : D \\to \\mathbb{R}\\)\uac00 \\(c\\)\ub97c \uc67c\ucabd \ub05d\uc810\uc73c\ub85c \ud558\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \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 \\in D\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x)-L| < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc6b0\uadf9\ud55c \\(L\\)\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0,\n\\[f(x) \\,\\to\\, L \\quad \\text{as} \\quad x \\,\\to\\, c^{+}\\]\n\ub610\ub294\n\\[\\lim_{x\\,\\to\\,c^{+}} f(x) = L\\]\n\ub610\ub294\n\\[f(c^+ ) = L\\]\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/li>\n<\/ol>\n<\/div>\n

    \uadf9\ud55c\uac12\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uacf3\uc5d0\uc11c\ub3c4 \uc88c\uadf9\ud55c\uc774\ub098 \uc6b0\uadf9\ud55c\uc740 \uc874\uc7ac\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \n

    \ubcf4\uae30 1.<\/span>
    \n\ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.
    \n\\[f(x) = \\begin{cases} 7 \\quad &\\text{if} \\,\\, x > 1 \\\\[8pt]
    \n0 \\quad &\\text{if} \\,\\, x \\le 1
    \n\\end{cases}\\]
    \n\uc774 \ub54c \\(1\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc9c0\ub9cc
    \n\\[\\lim_{x\\,\\to\\,1^-} f(x) = 0 ,\\quad \\lim_{x\\,\\to\\,1^+} f(x) = 7\\]
    \n\ub85c\uc11c \\(1\\)\uc5d0\uc11c \\(f\\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc740 \uac01\uac01 \uc874\uc7ac\ud55c\ub2e4.\n<\/p>\n<\/div>\n

    \ud568\uc218\uac00 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c \uad6c\uac04\uc758 \uc67c\ucabd \ub05d\uc810\uc5d0\uc11c\ub294 \uc6b0\uadf9\ud55c\uacfc \uadf9\ud55c\uc744 \ub3d9\uc77c\uc2dc\ud558\uba70, \uad6c\uac04\uc758 \uc624\ub978\ucabd \ub05d\uc810\uc5d0\uc11c\ub294 \uc88c\uadf9\ud55c\uacfc \uadf9\ud55c\uc744 \ub3d9\uc77c\uc2dc\ud55c\ub2e4.<\/p>\n

    \n

    \ubcf4\uae30 2.<\/span>
    \n\ud568\uc218 \\(f\\)\uac00 \ubc18\uc5f4\ub9b0 \uad6c\uac04 \\((3,\\,5]\\)\uc5d0\uc11c \\(f(x) = x^2\\)\uc73c\ub85c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \uc774 \ub54c
    \n\\[\\lim_{x\\,\\to\\,3} f(x) = \\lim_{x\\,\\to\\,3^+} f(x) = 9 ,\\]
    \n\\[\\lim_{x\\,\\to\\,5} f(x) = \\lim_{x\\,\\to\\,5^-} f(x) = 25 \\]
    \n\uc774\ub2e4. \ub610\ud55c \\(3 < c < 5\\)\uc778 \uc810 \\(c\\)\uc5d0 \ub300\ud574\uc11c\ub294\n\\[\\lim_{x\\,\\to\\,c} f(x) = \\lim_{x\\,\\to\\,c^-} f(x) = \\lim_{x\\,\\to\\,c^+} f(x) = c^2 \\]\n\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \ud55c\ubc29\ud5a5 \uadf9\ud55c\uacfc \uc591\ubc29\ud5a5 \uadf9\ud55c\uc758 \uad00\uacc4<\/h3>\n

    \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc744 \ud1b5\ud2c0\uc5b4 \ud55c\ubc29\ud5a5 \uadf9\ud55c\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud55c\ubc29\ud5a5 \uadf9\ud55c\uacfc \uad6c\ubd84\ud558\uae30 \uc704\ud558\uc5ec \uc77c\ubc18\uc801\uc778 \uadf9\ud55c\uc744 \uc591\ubc29\ud5a5 \uadf9\ud55c\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \ubcf4\uae30 2\uc5d0\uc11c \ubcf8 \uac83\ucc98\ub7fc \ud568\uc218\uc758 \uc815\uc758\uc5ed\uc774 \uad6c\uac04\uc77c \ub54c, \uad6c\uac04\uc758 \uc67c\ucabd \ub05d\uc810\uc5d0\uc11c\ub294 \uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \ub3d9\uc77c\ud55c \uac1c\ub150\uc774\uba70, \uad6c\uac04\uc758 \uc624\ub978\ucabd \ub05d\uc810\uc5d0\uc11c\ub294 \uadf9\ud55c\uacfc \uc88c\uadf9\ud55c\uc774 \ub3d9\uc77c\ud55c \uac1c\ub150\uc774\ub2e4.<\/p>\n

    \uadf8\ub7ec\ubbc0\ub85c \uc591\ubc29\ud5a5 \uadf9\ud55c\uc774\ub77c\ub294 \uc6a9\uc5b4\ub294 \\(f\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0\uad6c\uac04\uc758 \\(c\\)\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\uc744 \ub54c, \uc989 \\(x\\)\uac00 \\(c\\)\uc758 \uc67c\ucabd\uc73c\ub85c\ubd80\ud130 \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \uc218 \uc788\uace0, \\(c\\)\uc758 \uc624\ub978\ucabd\uc73c\ub85c\ubd80\ud130\ub3c4 \\(c\\)\ub85c \ub2e4\uac00\uac08 \uc218 \uc788\ub294 \uc0c1\ud669\uc5d0\uc11c\ub9cc \uc0ac\uc6a9\ud558\uae30\ub85c \ud55c\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 1.<\/span>
    \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\uc758 \\(c\\)\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \uc774 \ub54c
    \n\\[\\lim_{x\\,\\to\\,c} f(x) = L\\]
    \n\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740
    \n\\[\\lim_{x\\,\\to\\,c^-} f(x) = L \\quad \\text{and} \\quad \\lim_{x\\,\\to\\,c^+}f(x)=L\\]
    \n\uc778 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \uba3c\uc800 \ud544\uc694\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uac12\uc774 \\(L\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(0 < |x-c| < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ |f(x)-L| < \\epsilon\\tag{1}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370\n\\[0 < c-x < \\delta\\tag{2}\\]\n\ub610\ub294\n\\[0 < x-c < \\delta\\tag{3}\\]\n\uc5b4\ub290 \ub54c\ub098 \\(0 < |x-c| < \\delta\\)\uac00 \uc131\ub9bd\ud558\ubbc0\ub85c (1)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (2), (1)\uc5d0 \uc758\ud558\uc5ec\n\\[\\lim_{x\\,\\to\\,c^-} f(x) = L\\]\n\uc774\uba70, (3), (1)\uc5d0 \uc758\ud558\uc5ec\n\\[\\lim_{x\\,\\to\\,c^+} f(x) = L\\]\n\uc774\ub2e4.<\/p>\n

    \ub2e4\uc74c\uc73c\ub85c \ucda9\ubd84\uc870\uac74\uc744 \uc99d\uba85\ud558\uc790. \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \ubaa8\ub450 \\(L\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc88c\uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec
    \n\\[0 < c-x < \\delta_1\\tag{4}\\]\n\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x)-L| < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba70, \uc6b0\uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_2 > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec
    \n\\[0 < x-c < \\delta_2\\tag{5}\\]\n\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x) - L| < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \\(\\delta = \\min \\left\\{ \\delta_1 ,\\, \\delta_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(0 < |x-c| < \\delta\\)\uc778 \\(x\\)\ub294 (4) \ub610\ub294 (5) \uc911 \ud558\ub098\ub97c \ubc18\ub4dc\uc2dc \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c, \uadf8\ub7ec\ud55c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x) - L| < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span>\n<\/p>\n<\/div>\n

    \ud55c\ubc29\ud5a5 \uadf9\ud55c\uc758 \uc608: \\((\\sin x )\/ x\\)\uc758 \uadf9\ud55c<\/h3>\n

    \uc0bc\uac01\ud568\uc218\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uadf9\ud55c \uacf5\uc2dd\uc744 \ud558\ub098 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

    \n

    \uc815\ub9ac 2. (\\((\\sin \\theta )\/\\theta\\)\uc758 \uadf9\ud55c)<\/span>
    \n\\[\\lim_{\\theta \\to 0} \\frac{\\sin \\theta}{\\theta} = 1 \\qquad ( \\theta \\text{ in radians}) \\tag{6}\\]\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(\\theta =0\\)\uc5d0\uc11c \\((\\sin \\theta )\/\\theta \\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc758 \uac12\uc774 \ubaa8\ub450 \\(1\\)\uc784\uc744 \ubcf4\uc774\uc790. \uba3c\uc800 \\(0 < \\theta < \\pi \/2\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

    \"\"<\/p>\n

    \uadf8\ub9bc\uc640 \uac19\uc774 \uc88c\ud45c\ud3b4\uba74\uc758 \uc81c 1 \uc0ac\ubd84\uba74\uc5d0 \ubc18\uc9c0\ub984\uc774 \\(1\\)\uc774\uace0 \uc6d0\uc810\uc774 \uc911\uc2ec\uc778 \uc0ac\ubd84\uc6d0\uc744 \uadf8\ub9b0\ub2e4. \uadf8\ub9ac\uace0 \\(\\angle \\mathrm{POA} = \\theta\\)\uac00 \ub418\ub3c4\ub85d \uc0ac\ubd84\uc6d0 \uc704\uc758 \uc810 \\(\\mathrm{P}\\)\ub97c \uc7a1\ub294\ub2e4. \uc810 \\(\\mathrm{P}\\)\uc5d0\uc11c \\(x\\)\ucd95\uc5d0 \ub0b4\ub9b0 \uc218\uc120\uc758 \ubc1c\uc744 \\(\\mathrm{Q}\\)\ub77c\uace0 \ud558\uace0, \uc810 \\(\\mathrm{A}(1,\\,0)\\)\uc744 \uc9c0\ub098\uace0 \\(x\\)\ucd95\uacfc \uc218\uc9c1\uc778 \uc9c1\uc120\uc774 \ubc18\uc9c1\uc120 \\(\\mathrm{OP}\\)\uc640 \ub9cc\ub098\ub294 \uc810\uc744 \\(\\mathrm{T}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
    \n\\[\\sin \\theta = \\overline{\\mathrm{PQ}} ,\\quad \\cos \\theta = \\overline{\\mathrm{OQ}} ,\\quad \\tan \\theta = \\overline{\\mathrm{TA}}\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370
    \n\\[(\\text{Area } \\triangle \\mathrm{OAP}) \\le (\\text{Area sector } \\mathrm{OAP}) \\le (\\text{Area }\\triangle \\mathrm{OAT} )\\]
    \n\uc774\ubbc0\ub85c
    \n\\[\\frac{1}{2} \\sin \\theta \\le \\frac{1}{2}\\theta \\le \\frac{1}{2} \\tan \\theta \\]
    \n\uc774\ub2e4. \\(0 < \\theta < \\pi \/ 2\\)\uc758 \ubc94\uc704\uc5d0\uc11c \uc704 \ubd80\ub4f1\uc2dd\uc758 \uac01 \ud56d\uc740 \ubaa8\ub450 \uc591\uc218\uc774\ubbc0\ub85c, \uac01 \ud56d\uc5d0 \\(2\\)\ub97c \uacf1\ud558\uace0 \uac01 \ud56d\uc744 \\(\\sin \\theta\\)\ub85c \ub098\ub204\uba74\n\\[1 < \\frac{\\theta}{\\sin \\theta } < \\frac{1}{\\cos \\theta}\\]\n\uc774\uba70, \uc138 \ud56d\uc758 \uc5ed\uc218\ub97c \ucde8\ud558\uba74\n\\[1 > \\frac{\\sin\\theta}{\\theta} > \\cos\\theta \\]
    \n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7f0\ub370 \\(\\theta \\,\\to\\, 0^+\\)\uc77c \ub54c \\(\\cos \\theta \\,\\to\\,1\\)\uc774\ubbc0\ub85c, \uc704 \ubd80\ub4f1\uc2dd\uacfc \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
    \n\\[\\lim_{\\theta \\,\\to\\,0^+} \\frac{\\sin\\theta}{\\theta} = 1\\]
    \n\uc744 \uc5bb\ub294\ub2e4. \ud55c\ud3b8 \\((\\sin \\theta)\/\\theta\\)\ub294 \uc6b0\ud568\uc218\uc774\ubbc0\ub85c \\(0\\)\uc5d0\uc11c \uc774 \ud568\uc218\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc740 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
    \n\\[\\lim_{\\theta \\,\\to\\,0} \\frac{\\sin\\theta}{\\theta} = 1\\]
    \n\uc744 \uc5bb\ub294\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \ubcf4\uae30 3.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uac12\uc744 \uad6c\ud574 \ubcf4\uc790.
    \n\\[\\lim_{h\\to 0}\\frac{\\cos h -1}{h} \\]
    \n\\(\\cos h = 1 – 2 \\sin ^2 (h\/2)\\)\uc774\ubbc0\ub85c \uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec
    \n\\[\\begin{align}
    \n\\lim_{h\\to 0} \\frac{\\cos h -1}{h}
    \n&= \\lim_{h\\to 0}\\frac{- \\sin^2 (h\/2)}{h\/2} \\\\[6pt]
    \n&= -\\lim_{\\theta \\to 0} \\frac{\\sin \\theta}{\\theta} \\sin \\theta \\tag*{\\(\\leftarrow \\, \\theta = h\/2 \\)}\\\\[8pt]
    \n&= – 1 \\times 0 = 0.
    \n\\end{align}\\]\n<\/p>\n<\/div>\n

    \n

    \ubcf4\uae30 4.<\/span>
    \n\\(a \\ne 0,\\) \\(b \\ne 0\\)\uc77c \ub54c \ub2e4\uc74c \uadf9\ud55c\uac12\uc744 \uad6c\ud574 \ubcf4\uc790.
    \n\\[\\lim_{x\\,\\to\\,0} \\frac{\\sin ax}{bx}\\]
    \n\uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec
    \n\\[\\begin{align}
    \n\\lim_{x\\,\\to\\,0} \\frac{\\sin ax}{bx}
    \n&= \\lim_{x\\,\\to\\,0} \\frac{\\sin ax}{ax} \\cdot \\frac{a}{b} \\\\[6pt]
    \n&= \\frac{a}{b} \\cdot \\lim_{h\\,\\to\\,0} \\frac{\\sin h}{h} \\\\[8pt]
    \n&= \\frac{a}{b} \\cdot 1 = \\frac{a}{b}.
    \n\\end{align}\\]\n<\/p>\n<\/div>\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

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