{"id":1968,"date":"2019-04-05T12:12:50","date_gmt":"2019-04-05T03:12:50","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1968"},"modified":"2019-09-05T19:58:45","modified_gmt":"2019-09-05T10:58:45","slug":"calculus-indeterminate-forms-and-lhopitals-rule","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-indeterminate-forms-and-lhopitals-rule\/","title":{"rendered":"\ubd80\uc815\ud615 \uadf9\ud55c\uacfc \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59"},"content":{"rendered":"

\ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\ubaa8\ub450 \uc810 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0, \\(x\\to c\\)\uc77c \ub54c \\(f(x) \\to A,\\) \\(g(x) \\to B\\)\uc774\uba70 \\(B \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\lim_{x\\to c} \\frac{f(x)}{g(x)} = \\frac{A}{B}\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ub098 \ub9cc\uc57d \\(A = B = 0\\)\uc774\uac70\ub098, \\(x\\to c\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(g(x)\\)\uac00 \ubaa8\ub450 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uba74 \uc704\uc640 \uac19\uc740 \ub4f1\uc2dd\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \uc608\ucee8\ub300 \\(x\\to 0\\)\uc77c \ub54c \\(\\sin x \\to 0\\)\uc774\ubbc0\ub85c \uadf9\ud55c
\n\\[\\lim_{x\\to 0}\\frac{\\sin x}{x}\\]
\n\ub294 \uadf9\ud55c \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \uad6c\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n

\uc774\ucc98\ub7fc \uadf9\ud55c\uc774 \\(0\/0\\)\uaf34, \\(\\infty \/ \\infty\\)\uaf34, \\(\\infty \\cdot 0\\)\uaf34, \\((\\infty – \\infty)\\)\uaf34, \\(0^0\\)\uaf34, \\(1^\\infty\\)\uaf34, \\(\\infty^0\\)\uaf34\uc77c \ub54c\uc5d0\ub294 \uadf9\ud55c \uacc4\uc0b0 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \uadf9\ud55c\uac12\uc744 \uad6c\ud560 \uc218 \uc5c6\ub294\ub370, \uc774\ub7ec\ud55c \uadf9\ud55c\uc744 \ubd80\uc815\ud615<\/span>(indeterminate form) \uadf9\ud55c\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubd80\uc815\ud615 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud558\ub294 \uacf5\uc2dd\uc778 \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59<\/span>(L’H\u00f4pital’s rule)\uc744 \uc18c\uac1c\ud558\uace0, \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \uc5ec\ub7ec \uac00\uc9c0 \uc720\ud615\uc758 \ubd80\uc815\ud615 \uadf9\ud55c\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

<\/p>\n

\ubd80\uc815\ud615 \uadf9\ud55c\uc758 \uacc4\uc0b0: \\(0\/0\\)\uaf34<\/h3>\n

\ubd80\uc815\ud615 \uadf9\ud55c\uc758 \uc720\ud615\uc5d0 \ub530\ub77c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc758 \ub0b4\uc6a9\uc740 \uc870\uae08\uc529 \ub2e4\ub974\ub2e4. \uc5ec\uae30\uc11c\ub294 \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc744 \uc8fc\ub85c \uc0b4\ud3b4\ubcf4\uace0, \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc758 \uc99d\uba85\uc740 \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub05d\uc5d0\uc11c \ub2e4\ub8e8\uae30\ub85c \ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\ub85c\ud53c\ud0c8\uc758 \ubc95\uce59: \\(0\/0\\)\uaf34)<\/span><\/p>\n

\\(I\\)\uac00 \uad6c\uac04\uc774\uace0 \\(c\\)\uac00 \uc2e4\uc218\uc774\uba70 \ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790.<\/p>\n

    \n
  1. \\(c\\)\uac00 \\(I\\)\uc758 \ub0b4\uc810\uc774\uace0 \\(f\\)\uc640 \\(g\\)\uac00 \\(I\\setminus\\left\\{c\\right\\}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f(c)=g(c)=0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(x\\in I\\setminus\\left\\{ c\\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(g ‘ (x) \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d
    \n\\[\\lim_{x \\to c}\\frac{f ‘ (x)}{g ‘ (x)} = L\\]
    \n\uc774\uba74
    \n\\[\\lim_{x\\to c}\\frac{f(x)}{g(x)} = L\\]
    \n\uc774\ub2e4.<\/li>\n
  2. [1]\uc5d0\uc11c \\(I\\)\uac00 \ubc18\uc5f4\ub9b0 \uad6c\uac04 \\(I = (a,\\,c]\\)\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to c^-\\)\uc778 \uc88c\uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud558\uba70, \\(I\\)\uac00 \ubc18\ub2eb\ud78c \uad6c\uac04 \\(I = [c,\\,b)\\)\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to c^+\\)\uc778 \uc6b0\uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud55c\ub2e4.<\/li>\n
  3. [1], [2]\ub294 \\(L\\)\uc774 \uc2e4\uc218\uc77c \ub54c\uc5d0\ub3c4 \uc131\ub9bd\ud558\uace0 \\(L\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\uc774\uac70\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\uc77c \ub54c\uc5d0\ub3c4 \uc131\ub9bd\ud55c\ub2e4.<\/li>\n
  4. [3]\uc5d0\uc11c \\(I\\)\uac00 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uc9d1\ud569\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to \\infty\\)\uc778 \uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud558\uba70, \\(I\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uc9d1\ud569\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to -\\infty\\)\uc778 \uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

    \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc801\uc6a9\ud560 \ub550 \ubc18\ub4dc\uc2dc \\(x\\to c\\)\uc77c \ub54c \\(f ‘ (x) \/ g ‘ (x)\\)\uc758 \uadf9\ud55c\uc744 \uad6c\ud560 \uc218 \uc788\ub294\uc9c0 \uac80\ud1a0\ud574\uc57c \ud55c\ub2e4.<\/p>\n

    \n

    \uc608\uc81c 1.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uace0, \uc218\ub834\ud55c\ub2e4\uba74 \uadf9\ud55c\uac12\uc744 \uad6c\ud558\uc2dc\uc624.
    \n\\[\\lim_{x\\to 0}\\frac{5x-\\sin x}{x}.\\]\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(0\/0\\)\uaf34\uc778 \ubd80\uc815\ud615\uc774\ub2e4. \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
    \n\\[\\frac{5-\\cos x}{1}\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370
    \n\\[\\lim_{x\\to 0}\\frac{5-\\cos x}{1} = \\frac{5-1}{1} = 4\\]
    \n\uc774\ubbc0\ub85c, \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \uc218\ub834\ud558\uace0 \uadf9\ud55c\uac12\uc740
    \n\\[\\lim_{x\\to 0}\\frac{5x-\\sin x}{x}=4\\]
    \n\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \ub54c\uc5d0 \ub530\ub77c\uc11c\ub294 \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \ub450 \ubc88 \uc774\uc0c1 \uc0ac\uc6a9\ud574\uc57c \ud558\ub294 \uacbd\uc6b0\ub3c4 \uc788\ub2e4. \ub2e4\uc74c \uc608\uc81c\ub97c \ubcf4\ub77c.<\/p>\n

    \n

    \uc608\uc81c 2.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uace0, \uc218\ub834\ud55c\ub2e4\uba74 \uadf9\ud55c\uac12\uc744 \uad6c\ud558\uc2dc\uc624.
    \n\\[\\lim_{x\\to 0}\\frac{\\sqrt{1+x} -1 – \\frac{x}{2}}{x^2}.\\tag{1}\\]\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(0\/0\\)\uaf34\uc778 \ubd80\uc815\ud615\uc774\ub2e4. (1)\uc5d0\uc11c \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
    \n\\[\\lim_{x\\to 0}\\frac{\\frac{1}{2} (1+x)^{-1\/2} – \\frac{1}{2}}{2x}\\tag{2}\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc774 \uadf9\ud55c \ub610\ud55c \\(0\/0\\)\uaf34\uc778 \ubd80\uc815\ud615\uc774\ub2e4. (2)\uc5d0\uc11c \ub2e4\uc2dc \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
    \n\\[\\lim_{x\\to 0}\\frac{-\\frac{1}{4} (1+x)^{-3\/2}}{2}\\tag{3}\\]
    \n\uc774\uba70, \\(x=0\\)\uc744 \ub300\uc785\ud558\uc5ec \uadf9\ud55c\uac12\uc744 \uad6c\ud558\uba74 (3)\uc758 \uadf9\ud55c\uac12\uc740 \\(-\\frac{1}{8}\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec (2)\ub294 \uc218\ub834\ud558\uba70 \uadf8 \uadf9\ud55c\uac12\uc740 \\(-\\frac{1}{8}\\)\uc774\uace0, \ub2e4\uc2dc \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec (1)\ub3c4 \uc218\ub834\ud558\uba70 \uadf8 \uadf9\ud55c\uac12\uc740 \ub3d9\uc77c\ud558\uac8c \\(-\\frac{1}{8}\\)\uc774\ub2e4.<\/p>\n

    (Thomas\u2019 Calculus Global Edition 13\ud310 4.5\uc808\uc5d0\uc11c \ubc1c\ucdcc\ud568.)
    \n<\/span><\/p>\n<\/div>\n

    \\(f ‘ (x) \/ g ‘ (x)\\)\uac00 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud560 \ub54c\uc5d0\ub3c4 \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub2e4\uc74c \uc608\uc81c\ub97c \ubcf4\ub77c.<\/p>\n

    \n

    \uc608\uc81c 3.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.
    \n\\[\\lim_{x\\to 0^+} \\frac{\\sin x}{x^2}.\\]\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(0\/0\\)\uaf34\uc778 \ubd80\uc815\ud615\uc774\ub2e4. \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
    \n\\[\\lim_{x\\to 0^+} \\frac{\\cos x}{2x} = \\infty\\]
    \n\uc774\ubbc0\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
    \n\\[\\lim_{x\\to 0^+} \\frac{\\sin x}{x^2} = \\lim_{x\\to 0^+} \\frac{\\cos x}{2x} = \\infty\\]
    \n\uc774\ub2e4.\n<\/p>\n

    (Thomas\u2019 Calculus Global Edition 13\ud310 4.5\uc808\uc5d0\uc11c \ubc1c\ucdcc\ud568.)
    \n<\/span><\/p>\n<\/div>\n

    \n

    \uc8fc\uc758.<\/span>
    \n\ubd80\uc815\ud615\uc774 \uc544\ub2cc \uadf9\ud55c\uc744 \uacc4\uc0b0\ud560 \ub54c\uc5d0\ub294 \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uba74 \uc548 \ub41c\ub2e4. \uc608\ucee8\ub300
    \n\\[\\lim_{x\\to 0}\\frac{\\sin x}{1+3x} = \\lim_{x\\to 0}\\frac{\\cos x}{3} = \\frac{1}{3}\\]
    \n\uc774\ub77c\uace0 \ud558\uba74 \uc548 \ub41c\ub2e4. \uc65c\ub0d0\ud558\uba74 \uccab \uadf9\ud55c\uc774 \ubd80\uc815\ud615\uc774 \uc544\ub2c8\uae30 \ub54c\ubb38\uc774\ub2e4. \uc774 \uadf9\ud55c\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    <\/p>\n

    <\/p>\n

    \ubd80\uc815\ud615 \uadf9\ud55c\uc758 \uacc4\uc0b0: \\(\\infty \/ \\infty\\)\uaf34<\/h3>\n
    \n

    \uc815\ub9ac 2. (\ub85c\ud53c\ud0c8\uc758 \ubc95\uce59: \\(\\infty \/ \\infty\\)\uaf34)<\/span><\/p>\n

    \\(I\\)\uac00 \uad6c\uac04\uc774\uace0 \\(c\\)\uac00 \uc2e4\uc218\uc774\uba70 \ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\(I\\setminus\\left\\{c\\right\\}\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790.<\/p>\n

      \n
    1. \\(c\\)\uac00 \\(I\\)\uc758 \ub0b4\uc810\uc774\uace0 \\(f\\)\uc640 \\(g\\)\uac00 \\(I\\setminus\\left\\{ c \\right\\}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \\(x\\to c\\)\uc77c \ub54c \\(f(x) \\to \\infty,\\) \\(g(x) \\to \\infty\\)\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(x\\in I \\setminus\\left\\{ c\\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(g ‘ (x) \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d
      \n\\[\\lim_{x \\to c}\\frac{f ‘ (x)}{g ‘ (x)} = L\\]
      \n\uc774\uba74
      \n\\[\\lim_{x\\to c}\\frac{f(x)}{g(x)} = L\\]
      \n\uc774\ub2e4.<\/li>\n
    2. [1]\uc5d0\uc11c \\(x\\to c\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(g(x)\\)\uac00 \ubaa8\ub450 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uac70\ub098, \ub458 \uc911 \ud558\ub098\ub294 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uace0 \ub2e4\ub978 \ud558\ub098\ub294 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uc5ec\ub3c4 \uac19\uc740 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/li>\n
    3. [1]\uacfc [2]\uc5d0\uc11c \\(I\\)\uac00 \\(c\\)\ub97c \uc624\ub978\ucabd \ub05d\uc810\uc73c\ub85c \ud558\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to c^-\\)\uc778 \uc88c\uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud558\uba70, \\(I\\)\uac00 \\(c\\)\ub97c \uc67c\ucabd \ub05d\uc810\uc73c\ub85c \ud558\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to c^+\\)\uc778 \uc6b0\uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud55c\ub2e4.<\/li>\n
    4. [1]\uacfc [2]\uc5d0\uc11c \\(I\\)\uac00 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uc9d1\ud569\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to \\infty\\)\uc778 \uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud558\uba70, \\(I\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uc9d1\ud569\uc774\uace0 \uadf9\ud55c\uc774 \\(x\\to -\\infty\\)\uc778 \uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uadf8\ub300\ub85c \uc131\ub9bd\ud55c\ub2e4.<\/li>\n
    5. [1], [2], [3], [4]\ub294 \\(L\\)\uc774 \uc2e4\uc218\uc77c \ub54c\uc5d0\ub3c4 \uc131\ub9bd\ud558\uace0 \\(L\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\uc774\uac70\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\uc77c \ub54c\uc5d0\ub3c4 \uc131\ub9bd\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc608\uc81c 4.<\/span>
      \n\ub2e4\uc74c \uadf9\ud55c\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uace0, \uc218\ub834\ud55c\ub2e4\uba74 \uadf9\ud55c\uac12\uc744 \uad6c\ud558\uc2dc\uc624.
      \n\\[\\lim_{x\\to\\infty} \\frac{\\ln x}{\\sqrt{x}}.\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(\\infty\/\\infty\\)\uaf34\uc778 \ubd80\uc815\ud615\uc774\ub2e4. \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
      \n\\[\\lim_{x\\to \\infty}\\frac{1\/x}{1\/(2\\sqrt{x})} = \\lim_{x\\to\\infty}\\frac{1}{2\\sqrt{x}} = 0\\]
      \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \uc218\ub834\ud558\uace0
      \n\\[\\lim_{x\\to\\infty} \\frac{\\ln x}{\\sqrt{x}} = \\lim_{x\\to\\infty}\\frac{1}{2\\sqrt{x}} = 0\\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \ud65c\uc6a9\ud558\uc5ec \ud765\ubbf8\ub85c\uc6b4 \ubb38\uc81c\ub97c \ud480 \uc218 \uc788\ub2e4. \ub2e4\uc74c \uc608\uc81c\ub97c \ubcf4\ub77c.<\/p>\n

      \n

      \uc608\uc81c 5.<\/span>
      \n\ud568\uc218 \\(f\\)\uac00 \\((0,\\,\\infty)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0, \uc774 \uad6c\uac04\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\(x\\to\\infty\\)\uc77c \ub54c \\((f(x) + f ‘ (x))\\)\ub294 \uc218\ub834\ud558\uace0 \\(e^x f(x)\\)\ub294 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.
      \n\\[\\lim_{x\\to\\infty} f ‘ (x).\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\\(x\\to\\infty\\)\uc77c \ub54c \\(f(x)\\)\uc758 \uadf9\ud55c\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
      \n\\[\\lim_{x\\to\\infty} f(x) = \\lim_{x\\to\\infty}\\frac{e^x f(x)}{e^x}\\tag{4}\\]
      \n\uc774 \uadf9\ud55c\uc740 \\(\\infty \/ \\infty\\)\uaf34\uc778 \ubd80\uc815\ud615 \uadf9\ud55c\uc774\ub2e4. \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
      \n\\[\\lim_{x\\to\\infty}\\frac{e^x f(x) + e^x f ‘ (x)}{e^x} = \\lim_{x\\to\\infty} (f(x) + f ‘ (x))\\tag{5}\\]
      \n\uc774\uba70, \ubb38\uc81c\uc758 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec (5)\ub294 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec (4)\ub294 (5)\uc640 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud55c\ub2e4. (4)\uc758 \uadf9\ud55c\uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uba74
      \n\\[L = \\lim_{x\\to\\infty} f(x) = \\lim_{x\\to\\infty} (f(x) + f ‘ (x)) = L + \\lim_{x\\to\\infty} f ‘ (x)\\]
      \n\uc774\ubbc0\ub85c
      \n\\[\\lim_{x\\to\\infty} f ‘ (x) = 0\\]
      \n\uc774\ub2e4.
      \n(Wikipedia \ud398\uc774\uc9c0 L’H\u00f4pital’s rule<\/a>\uc5d0\uc11c \ubc1c\ucdcc\ud568, 2019\ub144 8\uc6d4 11\uc77c.)
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n

      \n

      \uc8fc\uc758.<\/span>
      \n\ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \uadf9\ud55c\uc744 \uacc4\uc0b0\ud560 \ub54c \ub4f1\uc2dd
      \n\\[\\lim_{x\\to c}\\frac{f(x)}{g(x)} = \\lim_{x\\to c}\\frac{f ‘ (x)}{g ‘ (x)}\\]
      \n\ub97c \uadf8\ub0e5 \uc0ac\uc6a9\ud558\uba74 \uc548 \ub41c\ub2e4. \uc774 \ub4f1\uc2dd\uc740 \uc6b0\ubcc0\uc758 \uadf9\ud55c\uc774 \uc9c4\ub3d9\ud558\uc9c0 \uc54a\uc744 \ub54c\uc5d0\ub9cc \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc608\ucee8\ub300
      \n\\[\\lim_{x\\to\\infty}\\frac{x+\\cos x}{x}\\]
      \n\ub294 \\(\\infty \/ \\infty\\)\uaf34\uc758 \ubd80\uc815\ud615 \uadf9\ud55c\uc774\uc9c0\ub9cc \u201c\ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\lim_{x\\to\\infty}\\frac{x+\\cos x}{x} = \\lim_{x\\to \\infty}\\frac{1-\\sin x}{1}\\]
      \n\uc774\ub2e4\u201d\ub77c\uace0 \uc4f0\uba74 \uc548 \ub41c\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc6b0\ubcc0\uc774 \uc9c4\ub3d9\ud558\uae30 \ub300\ubb38\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc8fc\uc758.<\/span>
      \n\ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0\uc11c \\(g ‘ (x) \\ne 0\\)\uc774\ub77c\ub294 \uc870\uac74\uc740 \ud544\uc218\uc774\ub2e4. \\(g ‘ (x) \\ne 0\\)\uc774\ub77c\ub294 \uc870\uac74\uc774 \ube60\uc9c0\uba74 \uc5b4\ub5a0\ud55c \ubb38\uc81c\uac00 \ubc1c\uc0dd\ud558\ub294\uc9c0 \uc608\ub97c \ud1b5\ud574 \uc0b4\ud3b4\ubcf4\uc790.
      \n\\[\\begin{align}
      \nf(x) &= x + \\sin x, \\\\[8pt]
      \ng(x) &= f(x) e^{\\sin x}
      \n\\end{align}\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[\\frac{f ‘ (x)}{g ‘ (x)} = \\frac{2 \\cos x \\cdot e^{-\\sin x}}{2 \\cos x + x + \\sin x \\cos x}\\]
      \n\uc774\ubbc0\ub85c \\(x\\to\\infty\\)\uc77c \ub54c \\(f ‘ (x) \/ g ‘ (x)\\)\ub294 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0\ub9cc \\(f(x)\/g(x)\\)\ub294 \uc9c4\ub3d9\ud55c\ub2e4.<\/p>\n

      (Wikipedia \ud398\uc774\uc9c0 L’H\u00f4pital’s rule<\/a>\uc5d0\uc11c \ubc1c\ucdcc\ud568, 2019\ub144 8\uc6d4 11\uc77c.)
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n

      \ubd80\uc815\ud615 \uadf9\ud55c\uc758 \uacc4\uc0b0: \uadf8 \ubc16\uc758 \uaf34<\/h3>\n

      \uc9c0\uae08\uae4c\uc9c0 \\(0\/0\\)\uaf34\uc758 \uadf9\ud55c\uacfc \\(\\infty \/ \\infty\\)\uaf34\uc758 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \ub2e4\ub978 \ubd80\uc815\ud615 \uadf9\ud55c\uc740 \\(0\/0\\)\uaf34\uc774\ub098 \\(\\infty \/ \\infty\\)\uaf34\ub85c \ubc14\uafbc \ub4a4 \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud55c\ub2e4. \ub2e4\uc74c \ub450 \uc608\uc81c(6, 7)\ub97c \ubcf4\ub77c.<\/p>\n

      \n

      \uc608\uc81c 6. (\\(\\infty \\cdot 0\\)\uaf34\uc758 \uadf9\ud55c)<\/span><\/p>\n

      \ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.
      \n\\[\\lim_{x\\to 0^+} x \\ln x.\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(\\infty \\cdot 0\\)\uaf34\uc774\ubbc0\ub85c \uc788\ub294 \uadf8\ub300\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \ubb38\uc81c\uc758 \uadf9\ud55c\uc744 \ubcc0\ud615\ud558\uba74
      \n\\[\\lim_{x\\to 0^+} x \\ln x = \\lim_{x\\to 0^+} \\frac{\\ln x}{1\/x}\\]
      \n\uc774\ub2e4. \uc774 \uadf9\ud55c\uc740 \\(\\infty \/ \\infty\\)\uaf34\uc758 \uadf9\ud55c\uc774\ub2e4. \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
      \n\\[\\lim_{x\\to 0^+}\\frac{1\/x}{-1\/(x^2 )} = \\lim_{x\\to 0^+} (-x) =0\\]
      \n\uc774\ubbc0\ub85c, \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\lim_{x\\to 0^+} x \\ln x = \\lim_{x\\to 0^+} (-x) =0\\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc608\uc81c 7. (\\((\\infty – \\infty)\\)\uaf34\uc758 \uadf9\ud55c)<\/span>
      \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.
      \n\\[\\lim_{x\\to 0^+} \\left( \\frac{1}{\\sin x} – \\frac{1}{x} \\right).\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\((\\infty – \\infty)\\)\uaf34\uc774\ubbc0\ub85c \uc788\ub294 \uadf8\ub300\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \ubb38\uc81c\uc758 \uadf9\ud55c\uc744 \ubcc0\ud615\ud558\uba74
      \n\\[\\lim_{x\\to 0^+} \\left( \\frac{1}{\\sin x} – \\frac{1}{x} \\right)
      \n= \\lim_{x\\to 0^+} \\frac{x-\\sin x}{x\\sin x}\\]
      \n\uc774\ub2e4. \uc774 \uadf9\ud55c\uc740 \\(0\/0\\)\uaf34\uc758 \uadf9\ud55c\uc774\ub2e4. \uadf9\ud55c\uc744 \uad6c\ud558\uace0\uc790 \ud558\ub294 \uc2dd\uc758 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
      \n\\[\\lim_{x\\to 0^+} \\frac{1-\\cos x}{\\sin x + x\\cos x}\\]
      \n\uc774\ub2e4. \uc774 \uadf9\ud55c\uc740 \ub610 \ub2e4\uc2dc \\(0\/0\\)\uaf34\uc758 \uadf9\ud55c\uc774\ub2e4. \ub2e4\uc2dc \ud55c \ubc88 \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
      \n\\[\\lim_{x\\to 0^+} \\frac{\\sin x}{2\\cos x – x\\sin x} = \\frac{0}{2} = 0\\]
      \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\begin{align}\\lim_{x\\to 0^+} \\left( \\frac{1}{\\sin x} – \\frac{1}{x} \\right)
      \n&= \\lim_{x\\to 0^+} \\frac{1-\\cos x}{\\sin x + x\\cos x}\\\\[6pt]
      \n&= \\lim_{x\\to 0^+} \\frac{\\sin x}{2\\cos x – x\\sin x} =0
      \n\\end{align}\\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \uc9c0\uc218\uc5d0 \ubcc0\uc218\uac00 \uc788\uc744 \ub54c\uc5d0\ub294 \ub85c\uadf8\ud568\uc218\ub97c \uc774\uc6a9\ud55c\ub2e4. \ub2e4\uc74c \uc138 \uc608\uc81c(8, 9, 10)\ub97c \ubcf4\ub77c.<\/p>\n

      \n

      \uc608\uc81c 8. (\\(0^0\\)\uaf34\uc758 \uadf9\ud55c)<\/span>
      \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.
      \n\\[\\lim_{x\\to 0^+} x^x.\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(0^0\\)\uaf34\uc774\ubbc0\ub85c \uc788\ub294 \uadf8\ub300\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n

      \\(f(x)=x^x,\\) \\(\\exp(x) = e^x\\)\uc774\ub77c\uace0 \ud558\uba74
      \n\\[\\lim_{x\\to 0^+} \\ln f(x) = \\lim_{x\\to 0^+} x \\ln x = \\lim_{x\\to 0^+} \\frac{\\ln x}{1\/x}\\]
      \n\uc774\ubbc0\ub85c \uc608\uc81c 6\uc758 \ubc29\ubc95\uc5d0 \ub530\ub77c \uc774 \uadf9\ud55c\uac12\uc740 \\(0\\)\uc774\ub2e4. \\(\\exp\\)\uac00 \\(0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c
      \n\\[\\begin{align}
      \n\\lim_{x\\to 0^+} f(x) &= \\lim_{x\\to 0^+} \\exp(\\ln f(x))\\\\[6pt]
      \n&= \\exp\\left( \\lim_{x\\to 0^+} f(x) \\right) = e^0 = 1
      \n\\end{align}\\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc608\uc81c 9. (\\(1^\\infty\\)\uaf34\uc758 \uadf9\ud55c)<\/span>
      \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.
      \n\\[\\lim_{x\\to 0^+} (1+x)^{1\/x}.\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(1^\\infty\\)\uaf34\uc774\ubbc0\ub85c \uc788\ub294 \uadf8\ub300\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n

      \\(f(x) = (1+x)^{1\/x},\\) \\(\\exp (x) = e^x\\)\uc774\ub77c\uace0 \ud558\uba74
      \n\\[\\lim_{x\\to 0^+} \\ln f(x)
      \n= \\lim_{x\\to 0^+} \\frac{\\ln (1+x)}{x}\\]
      \n\uc774\ub2e4. \uc774 \uadf9\ud55c\uc740 \\(\\infty \/ \\infty\\)\uaf34\uc758 \uadf9\ud55c\uc774\ub2e4. \uc6b0\ubcc0\uc758 \uadf9\ud55c\uc5d0\uc11c \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
      \n\\[\\lim_{x\\to 0^+}\\frac{1\/(1+x)}{1} = 1\\]
      \n\uc774\ubbc0\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\lim_{x\\to 0^+} \\ln f(x) = 1\\]
      \n\uc774\ub2e4. \\(\\exp\\)\uac00 \\(1\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c
      \n\\[\\begin{align}
      \n\\lim_{x\\to 0^+} f(x) &= \\lim_{x\\to 0^+} \\exp( \\ln f(x) )\\\\[6pt]
      \n&= \\lim_{x\\to 0^+} \\exp\\left( \\lim_{x\\to 0^+} \\ln f(x) \\right) \\\\[6pt]
      \n&= \\exp (1) = e
      \n\\end{align}\\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc608\uc81c 10. (\\(\\infty ^0\\)\uaf34\uc758 \uadf9\ud55c)<\/span>
      \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.
      \n\\[\\lim_{x\\to\\infty} x^{1\/x}.\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\ubb38\uc81c\uc758 \uadf9\ud55c\uc740 \\(\\infty ^0\\)\uaf34\uc774\ubbc0\ub85c \uc788\ub294 \uadf8\ub300\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n

      \\(f(x) = x^{1\/x},\\) \\(\\exp (x) = e^x\\)\uc774\ub77c\uace0 \ud558\uba74
      \n\\[\\lim_{x\\to\\infty} \\ln f(x) = \\lim_{x\\to\\infty} \\frac{\\ln x}{x}\\]
      \n\uc774\ub2e4. \uc774 \uadf9\ud55c\uc740 \\(\\infty \/ \\infty\\)\uaf34\uc758 \uadf9\ud55c\uc774\ub2e4. \uc6b0\ubcc0\uc758 \uadf9\ud55c\uc5d0\uc11c \ubd84\ubaa8\uc640 \ubd84\uc790\ub97c \uac01\uac01 \ubbf8\ubd84\ud558\uba74
      \n\\[\\lim_{x\\to \\infty} \\frac{1\/x}{1} = 0\\]
      \n\uc774\ubbc0\ub85c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\lim_{x\\to\\infty} \\ln f(x) = \\lim_{x\\to \\infty} \\frac{1\/x}{1} = 0\\]
      \n\uc774\ub2e4. \\(\\exp\\)\uac00 \\(0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c
      \n\\[\\begin{align}
      \n\\lim_{x\\to\\infty} f(x)
      \n&= \\lim_{x\\to\\infty} \\exp(\\ln f(x)) \\\\[6pt]
      \n&= \\exp\\left( \\lim_{x\\to\\infty} \\ln f(x) \\right) \\\\[6pt]
      \n&= \\exp(0) = 1
      \n\\end{align}\\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n

      \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc758 \uc99d\uba85<\/h3>\n

      \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc99d\uba85\ud558\uae30 \uc704\ud574\uc11c\ub294 \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac(Cauchy’s mean value theorem)\uac00 \ud544\uc694\ud558\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 3. (\ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

      \ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \uc784\uc758\uc758 \\(x\\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(g ‘ (x) \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[\\frac{f ‘ (c)}{g ‘ (c)} = \\frac{f(b)-f(a)}{g(b)-g(a)}\\]
      \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(g\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \ub77c\uadf8\ub791\uc8fc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
      \n\\[g ‘ (c) = \\frac{g(b) – g(a)}{b-a}\\]
      \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(g ‘ (c) \\ne 0\\)\uc774\ubbc0\ub85c \\(g(b)-g(a) \\ne 0\\)\uc774\ub2e4.<\/p>\n

      \uc774\uc81c \\([a,\\,b]\\)\uc5d0\uc11c \ud568\uc218 \\(F\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
      \n\\[F(x) = f(x) – f(a) – \\frac{f(b)-f(a)}{g(b)-g(a)} [g(x)-g(a)].\\]
      \n\uc5ec\uae30\uc11c \\(g(b)-g(a) \\ne 0\\)\uc774\ubbc0\ub85c \\(F\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc798 \uc815\uc758\ub418\uc5c8\uc73c\uba70, \\(F\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub530\ub77c\uc11c \ub77c\uadf8\ub791\uc8fc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
      \n\\[F ‘ (c) = f ‘ (c) – \\frac{f(b)-f(a)}{g(b)-g(a)} g ‘ (c) =0\\]
      \n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \\(g ‘ (c) \\ne 0\\)\uc774\ubbc0\ub85c \uc704 \uc2dd\uc744 \uc815\ub9ac\ud558\uba74
      \n\\[\\frac{f ‘ (c)}{g ‘ (c)} = \\frac{f(b)-f(a)}{g(b)-g(a)}\\]
      \n\ub97c \uc5bb\ub294\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \ucc38\uace0.<\/span>
      \n\ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\ub294 \ubb3c\uccb4\uc758 \uc6c0\uc9c1\uc784\uacfc \uad00\ub828\ud558\uc5ec \uadf8 \uc758\ubbf8\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \ub450 \ubb3c\uccb4 A, B\uac00 \uc218\uc9c1\uc120\uc0c1\uc5d0\uc11c \uc6c0\uc9c1\uc778\ub2e4\uace0 \uc0dd\uac01\ud558\uc790. \uc2dc\uc791\uc810\uacfc \ub05d\uc810\uc758 \uc704\uce58\ub294 \uc0c1\uad00 \uc5c6\ub2e4. \uc2dc\uac01 \\(t\\)\uc5d0 \ub300\ud55c A\uc758 \uc704\uce58\ub97c \\(f(t)\\)\ub77c \ud558\uace0, \uc2dc\uac01 \\(t\\)\uc5d0 \ub300\ud55c B\uc758 \uc704\uce58\ub97c \\(g(t)\\)\ub77c\uace0 \ud558\uc790. \uc2dc\uac01\uc774 \\(a=2\\)\ubd80\ud130 \\(b=5\\)\uae4c\uc9c0 \ubcc0\ud558\ub294 \ub3d9\uc548, \uc989 \uc2dc\uac01\uad6c\uac04 \\([a,\\,b] = [2,\\,5]\\)\uc5d0\uc11c \ub450 \ubb3c\uccb4\uc758 \ud3c9\uade0\uc18d\ub3c4\ub294 \uac01\uac01
      \n\\[\\frac{f(b)-f(a)}{b-a} = \\frac{f(5)-f(2)}{5-2},\\\\[8pt] \\frac{g(b)-g(a)}{b-a} = \\frac{g(5)-g(2)}{5-2}\\]
      \n\uc774\ub2e4. \ub9cc\uc57d
      \n\\[\\frac{\\frac{f(5)-f(2)}{5-2}}{\\frac{g(5)-g(2)}{5-2}} = \\frac{f(5)-f(2)}{g(5)-g(2)} = k = 3\\]
      \n\uc774\ub77c\uba74 \ub3d9\uc77c\ud55c \uc2dc\uac04 \ub3d9\uc548 \ubb3c\uccb4 A\uc758 \ud3c9\uade0\uc18d\ub3c4\ub294 \ubb3c\uccb4 B\uc758 \ud3c9\uade0\uc18d\ub3c4\uc758 \\(k=3\\)\ubc30\uc774\ub2e4. \uc774\ub54c A\uc758 \uc21c\uac04\uc18d\ub3c4\uac00 B\uc758 \uc21c\uac04\uc18d\ub3c4\uc758 \\(3\\)\ubc30\uac00 \ub418\ub294 \uc21c\uac04\uc774 \uc2dc\uac01\uad6c\uac04 \\((2,\\,5)\\) \uc548\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc989
      \n\\[\\frac{f ‘ (c)}{g ‘ (c)} = k=3\\]
      \n\uc774 \ub418\ub294 \\(c\\)\uac00 \\((2,\\,5)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4.
      \n<\/span>\n<\/p>\n<\/div>\n

      \uc774\uc81c \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc99d\uba85\ud558\uc790. \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc740 \uc5ec\ub7ec \uacbd\uc6b0\ub85c \ub098\ub204\uc5b4 \uc9c4\ub2e4. \uac01 \uacbd\uc6b0\ub97c \uc27d\uac8c \uae30\uc220\ud558\uae30 \uc704\ud558\uc5ec \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ud558\uae30\ub85c \ud558\uc790.<\/p>\n

      \n

      \n\\(x\\to c\\)\uc77c \ub54c \\(f(x) \\to A ,\\) \\(g(x) \\to B, \\) \\((f ‘ (x) \/ g ‘ (x))\\to L\\)\uc774\uace0 \\(f,\\) \\(g,\\) \\(c,\\) \\(A,\\) \\(B,\\) \\(L\\)\uc774 \uc815\ub9ac 1 \ub610\ub294 \uc815\ub9ac 2\uc758 \uac00\uc815\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74
      \n\\[\\lim_{x\\to c}\\frac{f(x)}{g(x)} = L\\]
      \n\uc774\ub2e4.\n<\/p>\n<\/div>\n

      (1) \\(\\boldsymbol{c\\in\\mathbb{R}},\\) \\(\\boldsymbol{A=B=0},\\) \\(\\boldsymbol{L\\in\\mathbb{R}}\\)\uc778 \uacbd\uc6b0\uc758 \uc99d\uba85<\/h4>\n

      \\(x\\to c^+\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(x\\to c^+\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(x\\to c^-\\)\uc778 \uacbd\uc6b0\uac00 \uc99d\uba85\ub418\uba70, \ub450 \uacbd\uc6b0\uc758 \uc99d\uba85\uc744 \uacb0\ud569\ud558\uba74 \\(x\\to c\\)\uc778 \uacbd\uc6b0\uc758 \uc99d\uba85\uc774 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

      \\(x\\)\uac00 \\(c\\)\ubcf4\ub2e4 \ud070 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(g ‘ (x) \\ne 0\\)\uc774\uba70, \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\frac{f ‘ (d)}{g ‘ (d)} = \\frac{f(x) – f(c)}{g(x)-g(c)}\\]
      \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(d\\)\uac00 \\(c\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(f(c)=g(c)=0\\)\uc774\ubbc0\ub85c \uc704 \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
      \n\\[\\frac{f ‘ (d)}{g ‘ (d)} = \\frac{f(x)}{g(x)}\\]
      \n\\(d\\)\ub294 \\(x\\)\uc640 \\(c\\) \uc0ac\uc774\uc758 \uc810\uc774\ubbc0\ub85c \\(x \\to c\\)\uc77c \ub54c \\(d \\to c\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
      \n\\[\\lim_{x\\to c^+} \\frac{f(x)}{g(x)} = \\lim_{d\\to c^+}\\frac{f ‘ (d)}{g ‘ (d)} = \\lim_{x\\to c^+}\\frac{f ‘ (x)}{g ‘ (x)}\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.
      \n<\/span><\/p>\n

      (2) \\(\\boldsymbol{c\\in\\mathbb{R}},\\) \\(\\boldsymbol{A=B=\\infty},\\) \\(\\boldsymbol{L\\in\\mathbb{R}}\\)\uc778 \uacbd\uc6b0\uc758 \uc99d\uba85<\/h4>\n

      \uc55e\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(x\\to c^+\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

      \\(\\left\\{x_n\\right\\}\\)\uc774 \\(c\\)\uc5d0 \uc218\ub834\ud558\uace0 \\(x_n > c\\)\uc774\uba70 \\(x_n \\in I\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uace0 \uc11c\ub85c \uac19\uc740 \uac12\uc744 \uac16\ub294 \ud56d\uc774 \uc5c6\ub294 \uc784\uc758\uc758 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc11c\ub85c \ub2e4\ub978 \uac01 \uc790\uc5f0\uc218 \\(k,\\) \\(n\\)\uc5d0 \ub300\ud558\uc5ec, \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\frac{f(x_k ) – f(x_n )}{g(x_k ) – g(x_n )} = \\frac{f ‘ (c_{k,n})}{g ‘ (c_{k,n})}\\]
      \n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(c_{k,n}\\)\uc774 \\(x_k\\)\uc640 \\(x_n\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c
      \n\\[\\begin{align}
      \n\\frac{f(x_n)}{g(x_n)} – \\frac{f(x_k)}{g(x_n)}
      \n&= \\frac{f(x_n) – f(x_k)}{g(x_n)}\\\\[4pt]
      \n&= \\frac{1}{g(x_n)} \\cdot (g(x_n ) – g(x_k )) \\cdot \\frac{f ‘ (c_{k,n})}{g ‘ (c_{k,n})} \\\\[4pt]
      \n&= \\left( 1-\\frac{g(x_k )}{g(x_n )} \\right) \\frac{f ‘ (c_{k,n})}{g ‘ (c_{k,n})}
      \n\\end{align}\\]
      \n\uc989
      \n\\[\\frac{f(x_n )}{g(x_n )} = \\frac{f(x_k )}{g(x_n )} – \\frac{g(x_k )}{g(x_n )} \\cdot \\frac{f ‘ (c_{k,n})}{g ‘ (c_{k,n})} + \\frac{f ‘ (c_{k,n})}{g ‘ (c_{k,n})}\\tag{6}\\]
      \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(A = \\infty\\)\uc774\ubbc0\ub85c \\(n\\to\\infty\\)\uc77c \ub54c \\(1\/g(x_n )\\to 0\\)\uc774\ub2e4. \uadf8\ub9ac\uace0 \\(c_{k,n}\\)\uc740 \\(x_k\\)\uc640 \\(x_n\\) \uc0ac\uc774\uc5d0 \uc788\uae30 \ub54c\ubb38\uc5d0 \\(k \\to \\infty,\\) \\(n\\to\\infty\\)\uc77c \ub54c \\(c_{k,n}\\to c\\)\uc774\ub2e4. \ub530\ub77c\uc11c (6)\uc5d0 \uc758\ud558\uc5ec \\(f(x_n ) \/ g(x_n ) \\to L\\)\uc774\ub2e4. \uc989 \\(0 < \\epsilon < 1\\)\uc778 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc790\uc5f0\uc218 \\(N_0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n\\ge N_0\\)\uc77c \ub54c\ub9c8\ub2e4\n\\[\\left\\lvert \\frac{f ' (c_{N_0 ,n} )}{g ' (c_{N_0 ,n} )} - L \\right\\rvert < \\frac{\\epsilon}{3}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(g(x_n)\\to\\infty\\)\uc774\ubbc0\ub85c \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n\\ge N_1\\)\uc77c \ub54c\ub9c8\ub2e4\n\\[\\left\\lvert \\frac{f(x_{N_0})}{g(x_n )}\\right\\rvert < \\frac{\\epsilon}{3}\\]\n\uadf8\ub9ac\uace0\n\\[\\left\\lvert \\frac{g(x_{N_0})}{g(x_n)}\\right\\rvert \\cdot \\left\\lvert \\frac{f ' (c_{N_0 ,n})}{g ' (c_{N_0 ,n})} \\right\\rvert < \\frac{\\epsilon}{3}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(N = \\max \\left\\{ N_0 ,\\, N_1 \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n\\ge N\\)\uc77c \ub54c\ub9c8\ub2e4\n\\[\\begin{align}\n\\left\\lvert \\frac{f(x_n)}{g(x_n)}-L \\right\\rvert\n&\\le\n\\left\\lvert \\frac{f(x_{N_0})}{g(x_n)} \\right\\rvert + \\left\\lvert \\frac{g(x_{N_0})}{g(x_n)} \\cdot \\frac{f ' (c_{N_0 ,n})}{g ' (c_{N_0 ,n})} \\right\\rvert \\\\[4pt]\n&\\quad + \\left\\lvert \\frac{f ' (c_{N_0 ,n})}{g ' (c_{N_0, n})} -L \\right\\rvert < \\epsilon\n\\end{align}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c\n\\[\\lim_{n\\to\\infty} \\frac{f(x_n)}{g(x_n)} = L\\]\n\uc744 \uc5bb\ub294\ub2e4. \uc5ec\uae30\uc11c \\(\\left\\{x_n\\right\\}\\)\uc740 \\(c\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc784\uc758\uc758 \uc218\uc5f4\uc774\ubbc0\ub85c\n\\[\\lim_{x\\to c} \\frac{f(x)}{g(x)} = L\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

      \\(A=B=-\\infty\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(f\\)\uc640 \\(g\\)\ub97c \\(-f\\)\uc640 \\(-g\\)\ub85c \ubc14\uafb8\uc5b4 \uc99d\uba85\ud558\uba74 \ub41c\ub2e4. \\(A=\\infty,\\) \\(B=-\\infty\\)\uc774\uac70\ub098 \\(A=-\\infty,\\) \\(B=\\infty\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(f\\)\uc640 \\(g\\) \uc911 \ud558\ub098\ub9cc \ubd80\ud638\ub97c \ubc14\uafb8\uc5b4 \uc99d\uba85\ud558\uba74 \ub41c\ub2e4.<\/span><\/p>\n

      (3) \\(\\boldsymbol{c=\\infty},\\) \\(\\boldsymbol{L\\in\\mathbb{R}}\\)\uc778 \uacbd\uc6b0\uc758 \uc99d\uba85<\/h4>\n

      \\((a,\\,\\infty )\\subseteq I\\)\uc778 \uc591\uc218 \\(a\\)\ub97c \ud0dd\ud558\uc790. \uac01 \\(y\\in (0,\\, 1\/a )\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[\\phi (y) := f\\left( \\frac{1}{y} \\right) ,\\,\\,\\, \\varphi (y) := g\\left( \\frac{1}{y} \\right)\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \uc5f0\uc1c4 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
      \n\\[\\frac{\\phi ‘ (y)}{\\varphi ‘ (y)}
      \n= \\frac{f ‘ (1\/y) (-1\/y^2)}{g ‘ (1\/y) (-1\/y^2 )}
      \n= \\frac{f ‘ (1\/y)}{g ‘ (1\/y)}
      \n\\]
      \n\uc744 \uc5bb\ub294\ub2e4. \uc5ec\uae30\uc11c \\(x = 1\/y\\in (a,\\,\\infty )\\)\uc774\ubbc0\ub85c
      \n\\[\\frac{f ‘ (x)}{g ‘ (x)} = \\frac{\\phi ‘ (y)}{\\varphi ‘ (y)}\\]
      \n\uc774\ub2e4. \uadf8\ub9ac\uace0 \\(x\\to\\infty\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(y = 1\/x \\to 0^+\\)\uc778 \uac83\uc774\ubbc0\ub85c, \\(c=0\\)\uacfc \\(I = (0,\\,1\/a)\\)\ub77c\uace0 \ub450\uc5c8\uc744 \ub54c \\(\\phi\\)\uc640 \\(\\varphi\\)\ub294 \uc55e\uc758 \ub450 \uacbd\uc6b0 (1) \ub610\ub294 (2)\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uac00 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
      \n\\[\\begin{align}
      \n\\lim_{x\\to\\infty}\\frac{f ‘ (x)}{g ‘ (x)}
      \n&= \\lim_{y\\to 0^+} \\frac{\\phi ‘ (y)}{\\varphi ‘ (y)} \\\\[4pt]
      \n&= \\lim_{y\\to 0^+} \\frac{\\phi (y)}{\\varphi (y)} \\\\[4pt]
      \n&= \\lim_{x\\to \\infty} \\frac{f(x)}{g(x)}
      \n\\end{align}\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4.
      \n<\/span><\/p>\n

      (4) \\(\\boldsymbol{L = \\infty}\\)\uc778 \uacbd\uc6b0\uc758 \uc99d\uba85<\/h4>\n

      \uc815\ub9ac\uc758 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(f\\)\uc640 \\(g\\)\ub294 \\(I\\setminus\\left\\{c\\right\\}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(x\\to c\\)\uc77c \ub54c \\(g(x)\/f(x)\\)\uc758 \uadf9\ud55c\uc740 \\(0\/0\\)\uaf34 \ub610\ub294 \\(\\infty \/ \\infty\\)\uaf34\uc774\uba70, \\(g ‘ (x) \/ f ‘ (x)\\)\uc758 \uadf9\ud55c\uc740 \\(0\\)\uc774\ub2e4. \ub354\uc6b1\uc774 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f ‘ (x) \\ne 0\\)\uc774\ub2e4. \ub9cc\uc57d \\(A=0\\)\uc774\uba74
      \n\\[f(x) = f(x) – f(c) = f ‘ (a)(x-c) \\ne 0\\]
      \n\uc774 \uc131\ub9bd\ud558\uace0, \ub9cc\uc57d \\(A=\\infty\\)\uc774\uba74 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ne 0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c0\uac70\ub098 \\(c\\)\ub97c \ub05d\uc810\uc73c\ub85c \ud558\ub294 \uad6c\uac04 \\(J\\subseteq I\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \uc784\uc758\uc758 \\(x\\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ne 0,\\) \\(f ‘ (x) \\ne 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774\ub85c\uc368 \\(f\\)\uc640 \\(g\\)\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uc5c8\uc744 \ub54c \uc55e\uc758 \uc138 \uacbd\uc6b0 (1), (2), (3) \uc911 \ud558\ub098\uc758 \uc870\uac74\uc774 \ubaa8\ub450 \ub9cc\uc871\ub418\ubbc0\ub85c
      \n\\[\\lim_{x\\to c}\\frac{g(x)}{f(x)} = \\lim_{x\\to c}\\frac{g ‘ (x)}{f ‘ (x)} = 0\\]
      \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(L = \\infty\\)\uc774\ubbc0\ub85c, \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \/ g(x) > 0\\)\uc774\ub2e4. \ud2b9\ud788 \\(x\\to c\\)\uc77c \ub54c \\(f(x) \/ g(x) \\to \\infty = L\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n<\/span><\/p>\n

      <\/p>\n

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

      \ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\ubaa8\ub450 \uc810 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0, \\(x\\to c\\)\uc77c \ub54c \\(f(x) \\to A,\\) \\(g(x) \\to B\\)\uc774\uba70 \\(B \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\lim_{x\\to c} \\frac{f(x)}{g(x)} = \\frac{A}{B}\\) \uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ub098 \ub9cc\uc57d \\(A = B = 0\\)\uc774\uac70\ub098, \\(x\\to c\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(g(x)\\)\uac00 \ubaa8\ub450 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uba74 \uc704\uc640 \uac19\uc740 \ub4f1\uc2dd\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \uc608\ucee8\ub300 \\(x\\to 0\\)\uc77c \ub54c \\(\\sin x \\to 0\\)\uc774\ubbc0\ub85c \uadf9\ud55c…<\/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":[269,266,267,265,264,268],"class_list":["post-1968","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-cauchys-mean-value-theorem","tag-indeterminate-form","tag-lhopitals-rule","tag-265","tag-264","tag-268"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1968","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=1968"}],"version-history":[{"count":83,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1968\/revisions"}],"predecessor-version":[{"id":3223,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1968\/revisions\/3223"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1968"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1968"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1968"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}