{"id":1936,"date":"2019-03-07T11:55:17","date_gmt":"2019-03-07T02:55:17","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1936"},"modified":"2019-09-05T15:07:39","modified_gmt":"2019-09-05T06:07:39","slug":"calculus-limits-involving-infinity","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-limits-involving-infinity\/","title":{"rendered":"\ubb34\ud55c\ub300\ub97c \ud3ec\ud568\ud55c \uadf9\ud55c"},"content":{"rendered":"

\ud568\uc218 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c \u2018\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \ubc1c\uc0b0<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

\ud568\uc218\uac00 \uc218\ub834\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0\ub97c \ubaa8\ub450 \ubc1c\uc0b0\uc774\ub77c\uace0\ub9cc \ud558\uae30\uc5d0\ub294 \uc544\uae4c\uc6b0\ubbc0\ub85c, \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0 \uc911\uc5d0\uc11c\ub3c4 \ud2b9\ubcc4\ud55c \uba87 \uac00\uc9c0 \uacbd\uc6b0\uc5d0 \ub300\ud574\uc11c\ub294 \uadf9\ud55c\uc744 \ub530\ub85c \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcfc \ud544\uc694\uac00 \uc788\ub2e4. \uc608\ucee8\ub300 \\(x\\)\uc758 \uac12\uc774 \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098 \ubb34\ud55c\ud788 \uc791\uc544\uc9c0\ub294 \uacbd\uc6b0\uc758 \uadf9\ud55c\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\uc73c\uba70, \\(x\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098 \ubb34\ud55c\ud788 \uc791\uc544\uc9c0\ub294 \uacbd\uc6b0\uc758 \uadf9\ud55c\ub3c4 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubb34\ud55c\ub300\uac00 \ud3ec\ud568\ub41c \uadf9\ud55c\uc744 \uc815\uc758\ud558\uace0, \uc774\ub4e4\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

<\/p>\n

\ubb34\ud55c\ub300\uc5d0\uc11c \uc810\uc5d0 \uc218\ub834\ud558\ub294 \uadf9\ud55c<\/h3>\n

\\(x\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098 \\(x\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \uc791\uc544\uc9c8 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \ud55c \uc810\uc5d0 \ub2e4\uac00\uac00\ub294 \uacbd\uc6b0\uc758 \uadf9\ud55c\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

\n

\uc815\uc758 1. (\ubb34\ud55c\ub300\uc5d0\uc11c \uc810\uc5d0 \uc218\ub834\ud558\ub294 \uadf9\ud55c)<\/span><\/p>\n

\\(f\\)\uac00 \ud568\uc218\uc774\uace0 \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n

    \n
  1. \\(f\\)\uc758 \uc815\uc758\uc5ed \\(I\\)\uac00 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x > M\\)\uc778 \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert f(x) – L \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \uc989\n\\[\\forall \\epsilon > 0 \\, \\exists M\\in\\mathbb{R} \\, \\forall x\\in I \\,:\\,\\, ( x > M \\,\\, \\rightarrow \\,\\, \\lvert f(x) – L \\rvert < \\epsilon )\\]\n\uc774\uba74, \u2018\\(x\\)\uac00 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \uac08 \ub54c \\(f(x)\\)\ub294 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
    \n\\[\\lim_{x\\to\\infty} f(x) = L\\]
    \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n
  2. \\(f\\)\uc758 \uc815\uc758\uc5ed \\(I\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x < N\\)\uc778 \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert f(x) - L \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \uc989\n\\[\\forall \\epsilon > 0 \\, \\exists N\\in\\mathbb{R} \\, \\forall x\\in I \\,:\\,\\, ( x < N \\,\\, \\rightarrow \\,\\, \\lvert f(x) - L \\rvert < \\epsilon )\\]\n\uc774\uba74, \u2018\\(x\\)\uac00 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \uac08 \ub54c \\(f(x)\\)\ub294 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
    \n\\[\\lim_{x\\to -\\infty} f(x) = L\\]
    \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

    \ud568\uc218 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc774 \uc704\ub85c \uc720\uacc4\uc77c \ub54c\uc5d0\ub294 \\(x \\to \\infty\\)\uc778 \uadf9\ud55c\uc740 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(f\\)\uc758 \uc815\uc758\uc5ed\uc774 \uc544\ub798\ub85c \uc720\uacc4\uc77c \ub54c\uc5d0\ub294 \\(x \\to -\\infty\\)\uc778 \uadf9\ud55c\uc740 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4. \uc608\ucee8\ub300 \ub2e4\uc74c \uadf9\ud55c\ub4e4\uc740 (\uc2e4\uc218 \ubc94\uc704\uc5d0\uc11c) \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4.
    \n\\[\\lim_{x\\to\\infty} \\sin^{-1} x ,\\\\[8pt]
    \n\\lim_{x\\to -\\infty} \\ln x .\\]
    \n\ubc18\uba74\uc5d0 \\(f(x)\\)\uac00 \uc815\uc758\ub418\uc9c0 \uc54a\ub294 \uc810 \\(x\\)\uac00 \ubb34\uc218\ud788 \ub9ce\ub354\ub77c\ub3c4, \\(f\\)\uc758 \uc815\uc758\uc5ed\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \\(x\\to\\infty\\)\uc778 \uadf9\ud55c\uc740 \uc815\uc758\ub420 \uc218 \uc788\uc73c\uba70, \\(f\\)\uc758 \uc815\uc758\uc5ed\uc774 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \\(x\\to\\infty\\)\uc778 \uadf9\ud55c\uc740 \uc815\uc758\ub420 \uc218 \uc788\ub2e4. \uc608\ucee8\ub300 \\(x = n \\pi ,\\, n \\in \\mathbb{Z}\\)\uc77c \ub54c
    \n\\[\\frac{\\sin x}{x^2 \\sin x}\\]
    \n\ub294 \uc815\uc758\ub418\uc9c0 \uc54a\uc9c0\ub9cc
    \n\\[\\lim_{x\\to\\infty} \\frac{\\sin x}{x^2 \\sin x} = 0\\]
    \n\uc774\ub2e4.<\/p>\n

    \n

    \n\ubcf4\uae30 1.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
    \n\\[\\lim_{x\\to \\infty} \\frac{1}{x} = 0 ,\\quad \\lim_{x\\to -\\infty} \\frac{1}{x} =0 .\\]
    \n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(M = \\frac{1}{\\epsilon}\\)\uc774\ub77c\uace0 \ud558\uba74 \\(x > M\\)\uc77c \ub54c
    \n\\[\\left\\lvert \\frac{1}{x} -0 \\right\\rvert = \\frac{1}{x} < \\frac{1}{M} = \\epsilon\\]\n\uc774\ubbc0\ub85c \uccab \ubc88\uc9f8 \uadf9\ud55c\uc774 \uc99d\uba85\ub41c\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(N = – \\frac{1}{\\epsilon}\\)\uc774\ub77c\uace0 \ud558\uba74 \\(x < N\\)\uc77c \ub54c\n\\[\\left\\lvert \\frac{1}{x} -0 \\right\\rvert = - \\frac{1}{x} < -\\frac{1}{N} = \\epsilon\\]\n\uc774\ubbc0\ub3c4 \ub450 \ubc88\uc9f8 \uadf9\ud55c\uc774 \uc99d\uba85\ub41c\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \n

    \n\ubcf4\uae30 2.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
    \n\\[\\lim_{x\\to\\infty} \\frac{2x+4}{x-3} = 2.\\]
    \n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uace0
    \n\\[M = \\frac{10}{\\epsilon} +3\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x > M\\)\uc778 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[x > \\frac{10}{\\epsilon} +3\\tag{1}\\]
    \n\uc774\uace0 \\(x > 3\\)\uc774\ubbc0\ub85c \uc704 \ubd80\ub4f1\uc2dd (1)\uc740
    \n\\[\\frac{10}{x-3} < \\epsilon\\]\n\uacfc \ub3d9\uce58\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x > M\\)\uc77c \ub54c
    \n\\[\\left\\lvert \\frac{2x+3}{x-3} -2 \\right\\rvert = \\frac{10}{x-3} < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \n

    \n\ubcf4\uae30 3.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
    \n\\[\\lim_{x\\to\\infty} \\frac{\\sin x}{x^2 \\sin x} = 0.\\]
    \n\ud568\uc218
    \n\\[f : \\, x \\,\\mapsto\\, \\frac{\\sin x}{x^2 \\sin x}\\]
    \n\ub294 \\(x \\ne n \\pi ,\\, n \\in \\mathbb{Z}\\)\uc77c \ub54c \uc815\uc758\ub418\ubbc0\ub85c, \uc774 \ud568\uc218\uc758 \uc815\uc758\uc5ed\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4.<\/p>\n

    \uc774\uc81c \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
    \n\\[M= \\max\\left\\{ 1,\\, \\frac{1}{\\epsilon} \\right\\}\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc5d0 \uc18d\ud558\ub294 \\(x\\) \uc911\uc5d0\uc11c \\(x > M\\)\uc778 \uac83\uc740 \ubaa8\ub450
    \n\\[\\begin{align}
    \n\\lvert f(x) – 0 \\rvert
    \n&= \\frac{\\sin x}{x^2 \\sin x} = \\frac{1}{x^2} \\\\[6pt]
    \n& \\le \\frac{1}{x}
    \n < \\frac{1}{M} \n \\le \\epsilon\n\\end{align}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\to\\infty\\)\uc77c \ub54c \\(f(x) \\to 0\\)\uc774\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \n

    \n\ubcf4\uae30 4.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
    \n\\[\\lim_{x\\to -\\infty} e^x = 0.\\]
    \n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uace0 \\(N = \\ln \\epsilon\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x < N\\)\uc77c \ub54c \\(x < \\ln \\epsilon\\)\uc774\ubbc0\ub85c\n\\[\\left\\lvert e^x -0 \\right\\rvert = e^x < e^{\\ln \\epsilon} = \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \uc810\uc5d0\uc11c \uc218\ub834\ud558\ub294 \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uc740 \\(x\\)\uac00 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \uac00\uac70\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \uac08 \ub54c \uc218\ub834\ud558\ub294 \ud568\uc218\uc758 \uadf9\ud55c\uc5d0\ub3c4 \uadf8\ub300\ub85c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \uc608\ucee8\ub300 \ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uc758 \uc815\uc758\uc5ed\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uace0 \\(x \\to \\infty\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(g(x)\\)\uac00 \uac01\uac01 \\(A,\\) \\(B\\)\uc5d0 \uc218\ub834\ud558\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \\[\\begin{align}
    \n\\lim_{x\\to\\infty} (f(x)+g(x)) &= A+B, \\\\[6pt]
    \n\\lim_{x\\to\\infty} (f(x)-g(x)) &= A-B, \\\\[6pt]
    \n\\lim_{x\\to\\infty} (f(x)g(x)) &= AB, \\\\[4pt]
    \n\\lim_{x\\to\\infty} \\frac{f(x)}{g(x)} &= \\frac{A}{B} \\,\\,\\,\\,(\\text{if }B \\ne 0).
    \n\\end{align}\\]<\/p>\n

    \ub610\ud55c \uc138 \ud568\uc218 \\(f,\\) \\(g,\\) \\(h\\)\uc758 \uc815\uc758\uc5ed\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uace0, \uc2e4\uc218 \\(X\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \\(x > X\\)\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[f(x) \\le g(x) \\le h(x)\\]
    \n\uc774\uba70 \\(x \\to \\infty\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(h(x)\\)\uac00 \uac19\uc740 \uac12 \\(L\\)\uc5d0 \uc218\ub834\ud558\uba74 \\(x \\to \\infty\\)\uc77c \ub54c \\(g(x)\\)\ub3c4 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc774 \uc815\ub9ac\ub97c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac<\/span> \ub610\ub294 \uc870\uc784 \uc815\ub9ac<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \n

    \n\ubcf4\uae30 5.<\/span>
    \n\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uba74
    \n\\[\\lim_{x\\to\\infty} \\frac{2x+4}{x-3}\\]
    \n\uc758 \uac12\uc744 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4.
    \n\\[\\begin{align}
    \n\\lim_{x\\to\\infty} \\frac{2x+4}{x-3}
    \n&= \\lim_{x\\to\\infty} \\frac{2+(4\/x)}{1-(3\/x)}\\\\[6pt]
    \n&= \\frac{\\lim_{x\\to\\infty} 2 + \\lim_{x\\to\\infty} (4\/x)}{\\lim_{x\\to\\infty} 1 – \\lim_{x\\to\\infty}(3\/x)} \\\\[6pt]
    \n&= \\frac{\\lim_{x\\to\\infty} 2 + 4 \\lim_{x\\to\\infty} (1\/x)}{\\lim_{x\\to\\infty} 1 – 3 \\lim_{x\\to\\infty}(1\/x)} \\\\[6pt]
    \n& = \\frac{2 + 4 \\times 0}{1 – 3\\times 0} = 2. \\tag*{\\(\\square\\)}
    \n\\end{align}\\]\n<\/p>\n<\/div>\n

    \n

    \n\ubcf4\uae30 6.<\/span>
    \n\ub2e4\uc74c \uadf9\ud55c\uac12\uc744 \uad6c\ud574 \ubcf4\uc790.
    \n\\[\\lim_{x\\to\\infty} \\frac{\\sin x}{x^2 \\sin x} \\]
    \n\ud568\uc218 \\(f,\\) \\(g\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
    \n\\[f(x) = \\frac{1}{x^2} ,\\,\\, g(x) = \\frac{\\sin x}{x^2 \\sin x}.\\]
    \n\uadf8\ub7ec\uba74 \\(f\\)\uc640 \\(g\\)\uc758 \uacf5\ud1b5\uc815\uc758\uc5ed\uc5d0 \uc788\ub294 \uc591\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[-f(x) \\le g(x) \\le f(x)\\]
    \n\uc774\uba70
    \n\\[\\lim_{x\\to\\infty} f(x) =0\\]
    \n\uc774\ubbc0\ub85c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
    \n\\[\\lim_{x\\to\\infty} g(x) =0\\]
    \n\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    <\/p>\n

    \uc810\uc5d0\uc11c \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uadf9\ud55c<\/h3>\n

    \\(x\\)\uc758 \uac12\uc774 \ud55c \uc810\uc5d0 \ub2e4\uac00\uac08 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098, \ubb34\ud55c\ud788 \uc791\uc544\uc9c0\ub294 \uacbd\uc6b0\uc758 \uadf9\ud55c\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \n

    \uc815\uc758 2. (\uc810\uc5d0\uc11c \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uadf9\ud55c)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790.<\/p>\n

      \n
    1. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < \\lvert x-c \\rvert < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989
      \n\\[\\forall B > 0 \\, \\exists \\delta > 0 \\, \\forall x \\in I \\,:\\,\\, ( 0 < \\lvert x-c \\rvert < \\delta \\,\\rightarrow \\, f(x) > B )\\]
      \n\uc774\uba74, \u2018\\(x\\)\uac00 \\(c\\)\uc5d0 \uc811\uadfc\ud560 \ub54c \\(f(x)\\)\ub294 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
      \n\\[\\lim_{x\\to c} f(x) = \\infty\\]
      \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n
    2. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < \\lvert x-c \\rvert < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) < -B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989\n\\[\\forall B > 0 \\, \\exists \\delta > 0 \\, \\forall x \\in I \\,:\\,\\, ( 0 < \\lvert x-c \\rvert < \\delta \\,\\rightarrow \\, f(x) < -B )\\]\n\uc774\uba74, \u2018\\(x\\)\uac00 \\(c\\)\uc5d0 \uc811\uadfc\ud560 \ub54c \\(f(x)\\)\ub294 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
      \n\\[\\lim_{x\\to c} f(x) = – \\infty\\]
      \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \ubcf4\uae30 7.<\/span>
      \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
      \n\\[\\lim_{x\\to 0} \\frac{1}{\\lvert x \\rvert} = \\infty .\\]
      \n\\(B > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\frac{1}{B}\\)\uc774\ub77c\uace0 \ud558\uba74 \\(0 < \\lvert x-0 \\rvert < \\delta\\)\uc77c \ub54c\n\\[ \\frac{1}{\\lvert x \\rvert} > \\frac{1}{\\delta} = B\\]
      \n\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \ubcf4\uae30 8.<\/span>
      \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
      \n\\[\\lim_{x\\to 1} \\frac{1-x}{(x-1)^3} = – \\infty .\\]
      \n\\(B > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = \\sqrt{1\/B}\\)\uc774\ub77c\uace0 \ud558\uba74 \\(0 < \\lvert x-1 \\rvert < \\delta\\)\uc77c \ub54c\n\\[\\begin{align}\n\\lvert x-1 \\rvert & < \\sqrt{\\frac{1}{B}} \\\\[6pt]\n(x-1)^2 & < \\frac{1}{B} \\\\[6pt]\n\\frac{1}{(x-1)^2} & > B
      \n\\end{align}\\]
      \n\uc774\ubbc0\ub85c
      \n\\[\\frac{1-x}{(x-1)^3} = \\frac{-1}{(x-1)^2} < -B \\]\n\uc774\ub2e4.\n<\/span><\/p>\n<\/div>\n

      \uc810\uc5d0\uc11c \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uadf9\ud55c\uc740 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\ub3c4 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

      \n

      \uc815\uc758 3. (\uc810\uc5d0\uc11c \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \ud55c \ubc29\ud5a5 \uadf9\ud55c)<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ud55c \ubc18\ub2eb\ud78c \uad6c\uac04 \\(I=[c,\\,d)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790.<\/p>\n

        \n
      1. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < x-c < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989
        \n\\[\\forall B > 0 \\, \\exists \\delta > 0 \\, \\forall x \\in I \\,:\\,\\, ( 0 < x-c < \\delta \\,\\rightarrow \\, f(x) > B )\\]
        \n\uc774\uba74, \u2018\\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc6b0\uadf9\ud55c\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
        \n\\[\\lim_{x\\to c^+} f(x) = \\infty\\]
        \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n
      2. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < x-c < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) < -B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989\n\\[\\forall B > 0 \\, \\exists \\delta > 0 \\, \\forall x \\in I \\,:\\,\\, ( 0 < x-c < \\delta \\,\\rightarrow \\, f(x) < -B )\\]\n\uc774\uba74, \u2018\\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc6b0\uadf9\ud55c\uc740 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
        \n\\[\\lim_{x\\to c^+} f(x) = -\\infty\\]
        \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n

        <\/p>\n

        \ub2e4\uc74c\uc73c\ub85c, \ud568\uc218 \\(g\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ud55c \ubc18\uc5f4\ub9b0 \uad6c\uac04 \\(J=(b,\\,c]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790.<\/p>\n

        <\/p>\n

      3. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < c-x < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(g(x) > B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989
        \n\\[\\forall B > 0 \\, \\exists \\delta > 0 \\, \\forall x \\in J \\,:\\,\\, ( 0 < c-x < \\delta \\,\\rightarrow \\, g(x) > B )\\]
        \n\uc774\uba74, \u2018\\(c\\)\uc5d0\uc11c \\(g\\)\uc758 \uc88c\uadf9\ud55c\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
        \n\\[\\lim_{x\\to c^-} g(x) = \\infty\\]
        \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n
      4. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(0 < c-x < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(g(x) < -B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989\n\\[\\forall B > 0 \\, \\exists \\delta > 0 \\, \\forall x \\in J \\,:\\,\\, ( 0 < c-x < \\delta \\,\\rightarrow \\, g(x) < -B )\\]\n\uc774\uba74, \u2018\\(c\\)\uc5d0\uc11c \\(g\\)\uc758 \uc88c\uadf9\ud55c\uc740 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
        \n\\[\\lim_{x\\to c^-} g(x) = -\\infty\\]
        \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
        \n

        \ubcf4\uae30 9.<\/span>
        \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
        \n\\[\\lim_{t\\to 0^+} \\ln t = -\\infty.\\]
        \n\\(B > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\delta = e^{-B}\\)\ub77c\uace0 \ud558\uba74 \\(\\delta > 0\\)\uc774\ub2e4. \uc774\uc81c \\(0 < t-0 < \\delta\\)\uc77c \ub54c\n\\[\\begin{align}\nt < \\delta &= e^{-B} ,\\\\[8pt]\ne^{\\ln t} &< e^{-B} ,\\\\[8pt]\n\\ln t &< -B\n\\end{align}\\]\n\uc774\ubbc0\ub85c \\(t\\to 0^+\\)\uc77c \ub54c \\(\\ln t \\to -\\infty\\)\uc774\ub2e4.\n<\/span><\/p>\n<\/div>\n

        \n

        \ubcf4\uae30 10.<\/span>
        \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
        \n\\[\\lim_{x\\to \\pi^-} \\frac{1}{\\sin x} = \\infty.\\]
        \n\ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc774\uc6a9\ud558\uc790.
        \n\\[\\sin(\\pi -x) = \\sin x .\\]
        \n\\(B > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\delta = \\min\\left\\{\\frac{1}{B},\\,\\frac{\\pi}{2}\\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(0 < \\pi -x < \\delta\\)\uc77c \ub54c\n\\[0 < \\sin(\\pi -x) < \\pi -x < \\delta\\]\n\uc774\ubbc0\ub85c\n\\[\n\\frac{1}{\\sin x} \n= \\frac{1}{\\sin (\\pi -x)}\n> \\frac{1}{\\delta}
        \n\\ge B
        \n\\]
        \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\to \\pi^-\\)\uc77c \ub54c \\(\\frac{1}{\\sin x} \\to \\infty\\)\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        <\/p>\n

        \ubb34\ud55c\ub300\uc5d0\uc11c \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uadf9\ud55c<\/h3>\n

        \\(x\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098 \ubb34\ud55c\ud788 \uc791\uc544\uc9c8 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098 \ubb34\ud55c\ud788 \uc791\uc544\uc9c0\ub294 \uacbd\uc6b0\uc758 \uadf9\ud55c\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

        \n

        \uc815\uc758 4. (\ubb34\ud55c\ub300\uc5d0\uc11c \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uadf9\ud55c)<\/span><\/p>\n

        \\(f\\)\uac00 \ud568\uc218\uc774\uace0 \\(f\\)\uc758 \uc815\uc758\uc5ed \\(I\\)\uac00 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790.<\/p>\n

          \n
        1. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x > M\\)\uc778 \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989
          \n\\[\\forall B > 0 \\, \\exists M \\in \\mathbb{R} \\, \\forall x \\in I \\,:\\,\\, ( x > M \\,\\rightarrow\\, f(x) > B )\\]
          \n\uc774\uba74, \u2018\\(x\\)\uac00 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \uac08 \ub54c \\(f(x)\\)\ub294 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
          \n\\[\\lim_{x\\to\\infty} f(x) = \\infty\\]
          \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/li>\n
        2. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x > M\\)\uc778 \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) < -B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989\n\\[\\forall B > 0 \\, \\exists M \\in \\mathbb{R} \\, \\forall x \\in I \\,:\\,\\, ( x > M \\,\\rightarrow\\, f(x) < -B )\\]\n\uc774\uba74, \u2018\\(x\\)\uac00 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \uac08 \ub54c \\(f(x)\\)\ub294 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
          \n\\[\\lim_{x\\to\\infty} f(x) = -\\infty\\]
          \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/li>\n

          \ub2e4\uc74c\uc73c\ub85c \\(g\\)\uac00 \ud568\uc218\uc774\uace0 \\(g\\)\uc758 \uc815\uc758\uc5ed \\(J\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790.<\/p>\n

        3. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x < N\\)\uc778 \ubaa8\ub4e0 \\(x\\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(g(x) > B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989
          \n\\[\\forall B > 0 \\, \\exists N \\in \\mathbb{R} \\, \\forall x \\in J \\,:\\,\\, ( x < N \\,\\rightarrow\\, g(x) > B )\\]
          \n\uc774\uba74, \u2018\\(x\\)\uac00 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \uac08 \ub54c \\(g(x)\\)\ub294 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
          \n\\[\\lim_{x\\to -\\infty} g(x) = \\infty\\]
          \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/li>\n
        4. \uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x < N\\)\uc778 \ubaa8\ub4e0 \\(x\\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(g(x) < -B\\)\uac00 \uc131\ub9bd\ud558\uba74, \uc989\n\\[\\forall B > 0 \\, \\exists N \\in \\mathbb{R} \\, \\forall x \\in J \\,:\\,\\, ( x < N \\,\\rightarrow\\, g(x) < -B )\\]\n\uc774\uba74, \u2018\\(x\\)\uac00 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \uac08 \ub54c \\(g(x)\\)\ub294 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294
          \n\\[\\lim_{x\\to -\\infty} g(x) = -\\infty\\]
          \n\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/li>\n<\/ol>\n<\/div>\n
          \n

          \n\ubcf4\uae30 11.<\/span>
          \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
          \n\\[\\lim_{x\\to\\infty} e^x = \\infty .\\]
          \n\\(B > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(M = \\ln B\\)\ub77c\uace0 \ud558\uba74 \\(x > M\\)\uc77c \ub54c \\(x > \\ln B\\)\uc774\ubbc0\ub85c
          \n\\[e^x > e^{\\ln B} = B\\]
          \n\uc774\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          \n

          \n\ubcf4\uae30 12.<\/span>
          \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.
          \n\\[\\lim_{x\\to -\\infty} \\frac{x^2 +1}{x+1} = -\\infty .\\]
          \n\\(B > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(N = \\min \\left\\{ -1 ,\\, -B+1 \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x < N\\)\uc77c \ub54c\n\\[\\frac{2}{x+1} <0\\]\n\uc774\uace0\n\\[x < -B+1\\]\n\uc774\ubbc0\ub85c\n\\[\\frac{x^2 +1}{x+1} = (x+1)+\\frac{2}{x+1} < x+1 < -B\\]\n\uc774\ub2e4.\n<\/span><\/p>\n<\/div>\n

          <\/p>\n

          \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc9c4\ub3d9<\/h3>\n

          \\(x \\to c\\)\uc778 \ud568\uc218 \\(f\\)\uc758 \uadf9\ud55c\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uace0, \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uc9c0 \uc54a\uace0, \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uc9c0\ub3c4 \uc54a\uc744 \ub54c \u2018\\(f\\)\uc758 \uadf9\ud55c\uc774 \uc9c4\ub3d9\ud55c\ub2e4<\/span>(oscillate)\u2019 \ub610\ub294 \u2018\\(x \\to c\\)\uc77c \ub54c \\(f\\)\ub294 \uc9c4\ub3d9\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

          <\/p>\n

          \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uc810\uadfc\uc120<\/h3>\n

          \uc9c1\uad00\uc801\uc73c\ub85c \ud568\uc218\uc758 \uadf8\ub798\ud504\uac00 \ud2b9\uc815\ud55c \uc9c1\uc120\uc5d0 \ud55c \uc5c6\uc774 \uac00\uae4c\uc6cc\uc9c8 \ub54c \uadf8 \uc9c1\uc120\uc744 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uc810\uadfc\uc120\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uae30\uc11c\ub294 \uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \uadf8\ub798\ud504\uc758 \uc810\uadfc\uc120\uc744 \uc815\ud655\ud558\uac8c \uc815\uc758\ud574 \ubcf4\uc790.<\/p>\n

          \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uc810\uadfc\uc120\uc740 \uc218\ud3c9\uc810\uadfc\uc120, \uc218\uc9c1\uc810\uadfc\uc120, \uc0ac\uc120\uc810\uadfc\uc120\uc73c\ub85c \ub098\ub20c \uc218 \uc788\ub2e4. \uba3c\uc800 \uc218\ud3c9\uc810\uadfc\uc120\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

          \n

          \uc815\uc758 5. (\uc218\ud3c9\uc810\uadfc\uc120)<\/span><\/p>\n

          \uc9c1\uc120 \\(y = b\\)\uac00 \ud568\uc218 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\ud3c9\uc810\uadfc\uc120<\/span>\uc774\ub77c \ud568\uc740 \ub2e4\uc74c \ub458 \uc911 \ud558\ub098 \uc774\uc0c1\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.
          \n\\[\\lim_{x\\to\\infty} f(x) = b \\quad\\text{or}\\quad \\lim_{x\\to -\\infty} f(x) = b.\\]\n<\/p>\n<\/div>\n

          \uc810\uadfc\uc120\uc740 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc640 \ub9cc\ub098\uc9c0 \uc54a\uc73c\uba74\uc11c \uac00\uae4c\uc6cc\uc9c0\uae30\ub3c4 \ud558\uc9c0\ub9cc, \ud568\uc218\uc758 \uadf8\ub798\ud504\uc640 \uad50\ucc28\ud558\uae30\ub3c4 \ud55c\ub2e4. \uc608\ucee8\ub300 \ud568\uc218
          \n\\[f(x) = \\frac{1}{x}\\]
          \n\uc758 \uadf8\ub798\ud504\ub294 \uc810\uadfc\uc120 \\(y=0\\)\uacfc \ub9cc\ub098\uc9c0 \uc54a\uc73c\uba70, \ud568\uc218
          \n\\[g(x) = \\frac{\\sin x}{x}\\]
          \n\uc758 \uadf8\ub798\ud504\ub294 \uc810\uadfc\uc120 \\(y=0\\)\uacfc \ubb34\uc218\ud788 \ub9ce\uc740 \uc810\uc5d0\uc11c \ub9cc\ub09c\ub2e4.<\/p>\n

          \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uc218\ud3c9\uc810\uadfc\uc120\uc744 \uad6c\ud560 \ub54c\uc5d0\ub294 \\(x\\to\\infty\\)\uc77c \ub54c\uc758 \uadf9\ud55c\uacfc \\(x\\to -\\infty\\)\uc77c \ub54c\uc758 \uadf9\ud55c\uc744 \uad6c\ud574\ubcf4\uba74 \ub41c\ub2e4.<\/p>\n

          \n

          \n\ubcf4\uae30 13.<\/span>
          \n\ud568\uc218 \\(f\\)\uac00
          \n\\[f(x) = \\frac{x-3}{\\lvert x \\rvert +1}\\]
          \n\uc73c\ub85c \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\ud3c9\uc810\uadfc\uc120\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\n

          \uba3c\uc800 \\(x > 0\\)\uc77c \ub54c
          \n\\[\\lim_{x\\to\\infty}\\frac{x-3}{\\lvert x \\rvert +1} = \\lim_{x\\to\\infty}\\frac{x-3}{x+1} = 1\\]
          \n\uc774\ubbc0\ub85c \uc9c1\uc120 \\(y=1\\)\uc774 \uc218\ud3c9\uc810\uadfc\uc120 \uc911 \ud558\ub098\uc774\ub2e4. \ub610\ud55c \\(x < 0\\)\uc77c \ub54c\n\\[\\lim_{x\\to -\\infty}\\frac{x-3}{\\lvert x \\rvert +1} = \\lim_{x\\to -\\infty}\\frac{x-3}{-x+1} = -1\\]\n\uc774\ubbc0\ub85c \uc9c1\uc120 \\(y=-1\\)\uc740 \ub610\ub2e4\ub978 \uc218\ud3c9\uc810\uadfc\uc120\uc774\ub2e4.<\/p>\n

          \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\ud3c9\uc810\uadfc\uc120\uc740 \\(y=1\\)\uacfc \\(y=-1\\)\uc774\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          \ub2e4\uc74c\uc73c\ub85c \uadf8\ub798\ud504\uc758 \uc218\uc9c1\uc810\uadfc\uc120\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

          \n

          \uc815\uc758 6. (\uc218\uc9c1\uc810\uadfc\uc120)<\/span><\/p>\n

          \uc9c1\uc120 \\(x = a\\)\uac00 \ud568\uc218 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\uc9c1\uc810\uadfc\uc120<\/span>\uc774\ub77c \ud568\uc740 \ub2e4\uc74c \ub137 \uc911 \ud558\ub098 \uc774\uc0c1\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.
          \n\\[\\begin{align}
          \n\\lim _ { x\\to a^+} f(x) = \\infty &, \\quad
          \n\\lim _ { x\\to a^+} f(x) = -\\infty , \\\\[6pt]
          \n\\lim _ { x\\to a^-} f(x) = \\infty &\\quad \\text{or}\\quad
          \n\\lim _ { x\\to a^-} f(x) = -\\infty .
          \n\\end{align}\\]\n<\/p>\n<\/div>\n

          \ubd84\uc218\uc2dd\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uc218\uc9c1\uc810\uadfc\uc120\uc744 \uad6c\ud560 \ub54c\uc5d0\ub294 \ubd84\uc218\uc2dd\uc758 \ubd84\ubaa8\uac00 \\(0\\)\uc774 \ub418\ub294 \uc810\uc744 \ucc3e\uace0, \uadf8 \uc810\uc5d0\uc11c \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc744 \uad6c\ud574\ubcf4\uba74 \ub41c\ub2e4.<\/p>\n

          \n

          \n\ubcf4\uae30 14.<\/span>
          \n\ud568\uc218 \\(f\\)\uac00
          \n\\[f(x) = \\frac{x-3}{x^3 +2x^2 -5x -6}\\]
          \n\uc73c\ub85c \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\uc9c1\uc810\uadfc\uc120\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\n

          \uc6b0\ubcc0\uc758 \ubd84\uc218\uc2dd\uc758 \ubd84\ubaa8\ub97c \uc778\uc218\ubd84\ud574\ud558\uba74
          \n\\[f(x) = \\frac{x-3}{(x-1)(x-2)(x-3)}\\]
          \n\uc774\ubbc0\ub85c \\(x=1,\\) \\(x=2,\\) \\(x=3\\)\uc77c \ub54c \ubd84\uc218\uc2dd\uc758 \ubd84\ubaa8\uac00 \\(0\\)\uc774 \ub41c\ub2e4.<\/p>\n

          \\(x \\to 1^{\\pm}\\)\uc77c \ub54c \\(f(x) \\to \\mp \\infty\\)\uc774\ubbc0\ub85c(\ubcf5\ubd80\ud638\ub3d9\uc21c), \uc9c1\uc120 \\(x=1\\)\uc740 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\uc9c1\uc810\uadfc\uc120 \uc911 \ud558\ub098\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(x \\to 2^{\\pm}\\)\uc77c \ub54c \\(f(x) \\to \\pm \\infty\\)\uc774\ubbc0\ub85c(\ubcf5\ubd80\ud638\ub3d9\uc21c), \uc9c1\uc120 \\(x=2\\)\ub3c4 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\uc9c1\uc810\uadfc\uc120\uc774\ub2e4. \ud55c\ud3b8 \\(x\\to 3\\)\uc77c \ub54c\uc5d0\ub294 \\(f(x) \\to \\frac{1}{2}\\)\ub85c\uc11c \\(f\\)\uac00 \uc218\ub834\ud558\ubbc0\ub85c \\(x=3\\)\uc740 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc810\uadfc\uc120\uc774 \uc544\ub2c8\ub2e4.<\/p>\n

          \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc218\uc9c1\uc810\uadfc\uc120\uc740 \\(x=1,\\) \\(x=2\\)\uc774\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          \ub9c8\uc9c0\ub9c9\uc73c\ub85c \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

          \n

          \uc815\uc758 7. (\uc0ac\uc120\uc810\uadfc\uc120)<\/span><\/p>\n

          \uc9c1\uc120 \\(y = ax+b\\)\uac00 \ud568\uc218 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120<\/span>\uc774\ub77c \ud568\uc740 \ub2e4\uc74c \ub458 \uc911 \ud558\ub098 \uc774\uc0c1\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.
          \n\\[\\lim_{x\\to\\infty} [f(x) – (ax+b)] = 0 \\quad\\text{or}\\quad \\lim_{x\\to -\\infty} [f(x)-(ax+b)] = 0.\\]\n<\/p>\n<\/div>\n

          \uc0ac\uc120\uc810\uadfc\uc120\uc758 \uc815\uc758\uc5d0\uc11c \ubcf4\ub2e4\uc2dc\ud53c \uc0ac\uc120\uc810\uadfc\uc120\uc758 \ubc29\uc815\uc2dd \\(y=ax+b\\)\ub97c \uad6c\ud558\ub824\uba74 \uae30\uc6b8\uae30 \\(a\\)\uc640 \\(y\\)\uc808\ud3b8 \\(b\\)\ub97c \uad6c\ud574\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadf9\ud55c\uac12\uc744 \ud55c \ubc88\ub9cc \uad6c\ud574\uc11c\ub294 \uc0ac\uc120\uc810\uadfc\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \uad6c\ud560 \uc218 \uc5c6\ub2e4. \ub300\uc2e0 \uadf9\ud55c\uac12\uc744 \ub450 \ubc88 \uad6c\ud558\uc5ec \uc0ac\uc120\uc810\uadfc\uc120\uc758 \uae30\uc6b8\uae30\uc640 \\(y\\)\uc808\ud3b8\uc744 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n

          \uc989 \ud568\uc218 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \uad6c\ud558\ub294 \uacfc\uc815\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

            \n
          • \uba3c\uc800 \ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud55c\ub2e4.\\[\\lim_{x\\to\\infty}\\frac{f(x)}{x}\\]\uc774 \uadf9\ud55c\uc774 \\(0\\)\uc774 \uc544\ub2cc \uac12 \\(a\\)\uc5d0 \uc218\ub834\ud558\uba74 \\(a\\)\ub294 \uc0ac\uc120\uc810\uadfc\uc120\uc758 \uae30\uc6b8\uae30\uac00 \ub420 \uc218 \uc788\ub294 \ud6c4\ubcf4\uac12\uc774\ub2e4. \ub9cc\uc57d \uc774 \uadf9\ud55c\uc774 \uc591\uc758 \ubb34\ud55c\ub300 \ub610\ub294 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uac70\ub098 \\(0\\)\uc5d0 \uc218\ub834\ud558\uba74 \\(x\\to\\infty\\)\uc77c \ub54c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n
          • \uc774\uc81c \ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud55c\ub2e4.\\[\\lim_{x\\to\\infty}(f(x) – ax)\\]\uc774 \uadf9\ud55c\uc774 \\(b\\)\uc5d0 \uc218\ub834\ud558\uba74 \uc0ac\uc120\uc810\uadfc\uc120\uc758 \ubc29\uc815\uc2dd\uc740 \\(y=ax+b\\)\uac00 \ub41c\ub2e4. \ub9cc\uc57d \uc774 \uadf9\ud55c\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \\(x\\to\\infty\\)\uc77c \ub54c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n
          • \\(x\\to -\\infty\\)\uc77c \ub54c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120\uc744 \uad6c\ud558\ub294 \uacfc\uc815\ub3c4 \uc704\uc640 \ub3d9\uc77c\ud558\ub2e4.<\/li>\n<\/ul>\n
            \n

            \n\ubcf4\uae30 15.<\/span>
            \n\ud568\uc218 \\(f\\)\uac00
            \n\\[f(x) = \\frac{x^2-3}{\\lvert x \\rvert +1}\\]
            \n\uc73c\ub85c \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\n

            \uba3c\uc800
            \n\\[\\lim_{x\\to\\infty}\\frac{f(x)}{x} = 1\\]
            \n\uc774\uace0
            \n\\[\\lim_{x\\to\\infty}(f(x)-x) = -1\\]
            \n\uc774\ubbc0\ub85c \\(y=x-1\\)\uc740 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120 \uc911 \ud558\ub098\uc774\ub2e4.<\/p>\n

            \ub2e4\uc74c\uc73c\ub85c
            \n\\[\\lim_{x\\to -\\infty}\\frac{f(x)}{x} = -1\\]
            \n\uc774\uace0
            \n\\[\\lim_{x\\to -\\infty}(f(x)-(-x)) = -1\\]
            \n\uc774\ubbc0\ub85c \\(y=-x-1\\)\uc740 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ub610\ub2e4\ub978 \uc0ac\uc120\uc810\uadfc\uc120\uc774\ub2e4.<\/p>\n

            \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc0ac\uc120\uc810\uadfc\uc120\uc740 \\(y=x-1,\\) \\(y=-x-1\\)\uc774\ub2e4.
            \n<\/span><\/p>\n<\/div>\n

            <\/p>\n

            <\/h3>\n

            <\/p>\n

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

            \ud568\uc218 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c \u2018\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \ubc1c\uc0b0\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ud568\uc218\uac00 \uc218\ub834\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0\ub97c \ubaa8\ub450 \ubc1c\uc0b0\uc774\ub77c\uace0\ub9cc \ud558\uae30\uc5d0\ub294 \uc544\uae4c\uc6b0\ubbc0\ub85c, \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0 \uc911\uc5d0\uc11c\ub3c4 \ud2b9\ubcc4\ud55c \uba87 \uac00\uc9c0 \uacbd\uc6b0\uc5d0 \ub300\ud574\uc11c\ub294 \uadf9\ud55c\uc744 \ub530\ub85c \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcfc \ud544\uc694\uac00 \uc788\ub2e4. \uc608\ucee8\ub300 \\(x\\)\uc758 \uac12\uc774 \\(c\\)\uc5d0 \ub2e4\uac00\uac08 \ub54c \\(f(x)\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098 \ubb34\ud55c\ud788 \uc791\uc544\uc9c0\ub294 \uacbd\uc6b0\uc758 \uadf9\ud55c\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\uc73c\uba70, \\(x\\)\uc758 \uac12\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c0\uac70\ub098 \ubb34\ud55c\ud788 \uc791\uc544\uc9c0\ub294 \uacbd\uc6b0\uc758 \uadf9\ud55c\ub3c4 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294…<\/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":[166,140,141,164,142,163,138,137,165,139],"class_list":["post-1936","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-asymptote","tag-convergence","tag-divergence","tag-infinity","tag-oscillation","tag-163","tag-138","tag-137","tag-165","tag-139"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1936","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=1936"}],"version-history":[{"count":87,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1936\/revisions"}],"predecessor-version":[{"id":3229,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1936\/revisions\/3229"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1936"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1936"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1936"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}