{"id":6671,"date":"2021-07-20T23:48:52","date_gmt":"2021-07-20T14:48:52","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6671"},"modified":"2026-09-27T17:10:04","modified_gmt":"2026-09-27T08:10:04","slug":"calculating-limits","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/calculating-limits\/","title":{"rendered":"\uc218\uc5f4\uc758 \uadf9\ud55c \uacf5\uc2dd"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 1\uc7a5 3\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc758 \uadf9\ud55c\uc744 \uad6c\ud560 \ub54c\ub9c8\ub2e4 \\(\\epsilon\\)-\\(N\\) \uc815\uc758\ub97c \uc9c1\uc811 \uc0ac\uc6a9\ud558\ub294 \uac83\uc740 \ubc88\uac70\ub86d\ub2e4. \uc774 \uc808\uc5d0\uc11c\ub294 \uc774\ubbf8 \uc54c\uace0 \uc788\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc73c\ub85c\ubd80\ud130 \uc0c8\ub85c\uc6b4 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\uac8c \ud574 \uc8fc\ub294 \uae30\ubcf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

<\/p>\n

\uc0ac\uce59\uc5f0\uc0b0\uacfc \uad00\ub828\ub41c \uadf9\ud55c\uc758 \uc131\uc9c8<\/h2>\n
\n

\uc815\ub9ac 1.3.1.<\/span>
\n\\(\\left\\{a_n\\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\uace0 \\(A\\)\uc640 \\(B\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
\n\\[
\n\\lim_{n\\rightarrow\\infty}a_n=A
\n\\quad\\text{and}\\quad
\n\\lim_{n\\rightarrow\\infty}b_n=B
\n\\]
\n\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \n
  1. \\(\\displaystyle \\lim_{n\\rightarrow\\infty}(ka_n)=kA.\\)   (\\(k\\)\ub294 \uc2e4\uc218\uc778 \uc0c1\uc218.)<\/li>\n
  2. \\(\\displaystyle \\lim_{n\\rightarrow\\infty}(a_n+b_n)=A+B.\\)<\/li>\n
  3. \\(\\displaystyle \\lim_{n\\rightarrow\\infty}(a_n-b_n)=A-B.\\)<\/li>\n
  4. \\(\\displaystyle \\lim_{n\\rightarrow\\infty}(a_nb_n)=AB.\\)<\/li>\n
  5. \\(B\\ne0\\)\uc774\uba74 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(b_n\\ne0\\)\uc774\uace0,
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}\\frac{a_n}{b_n}
    \n=
    \n\\frac{A}{B}.
    \n\\]\n<\/li>\n
  6. \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uba74
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}(a_n)^m=A^m.
    \n\\]\n<\/li>\n
  7. \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(A\\ge0\\)\uc774\uba70 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\ge0\\)\uc774\uba74
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}\\sqrt[m]{a_n}
    \n=
    \n\\sqrt[m]{A}.
    \n\\]\n<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uc99d\uba85.<\/p>\n

    (1) \\(k=0\\)\uc774\uba74 \uc790\uba85\ud558\ub2e4. \\(k\\ne0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc591\uc218 \\(\\epsilon\\)\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(a_n\\rightarrow A\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|a_n-A| < \\frac{\\epsilon}{|k|}\n\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c\n\\[\n|ka_n-kA|\n=\n|k||a_n-A|\n< \\epsilon.\n\\]\n\uadf8\ub7ec\ubbc0\ub85c \\(ka_n\\rightarrow kA\\)\uc774\ub2e4.<\/p>\n

    (2) \uc591\uc218 \\(\\epsilon\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|a_n-A| < \\frac{\\epsilon}{2},\n\\qquad\n|b_n-B| < \\frac{\\epsilon}{2}\n\\]\n\uac00 \ub3d9\uc2dc\uc5d0 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\uba74 \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec\n\\[\n\\begin{aligned}\n|(a_n+b_n)-(A+B)|\n&\\le\n|a_n-A|+|b_n-B|\\\\\n&< \\epsilon.\n\\end{aligned}\n\\]\n\ub530\ub77c\uc11c \\(a_n+b_n\\rightarrow A+B\\)\uc774\ub2e4.<\/p>\n

    (3) (1)\uacfc (2)\ub97c \uc774\uc6a9\ud558\uc5ec
    \n\\[
    \na_n-b_n=a_n+(-1)b_n
    \n\\]
    \n\ub85c \ub193\uc73c\uba74 \ubc14\ub85c \uc5bb\uc5b4\uc9c4\ub2e4.<\/p>\n

    (4) \uba3c\uc800 \\(a_n\\rightarrow A\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|a_n-A| < 1\n\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \uadf8\ub7ec\ud55c \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n|a_n|\n\\le\n|A|+1.\n\\]\n\\(M=|A|+1\\)\uc774\ub77c\uace0 \ud558\uc790. \uc591\uc218 \\(\\epsilon\\)\uc774 \uc8fc\uc5b4\uc9c0\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n|b_n-B| < \\frac{\\epsilon}{2M}\n\\]\n\uc774\uace0\n\\[\n|a_n-A|\n<\n\\frac{\\epsilon}{2(|B|+1)}\n\\]\n\uc774 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74\n\\[\n\\begin{aligned}\n|a_nb_n-AB|\n&=\n|a_n(b_n-B)+B(a_n-A)|\\\\\n&\\le\n|a_n||b_n-B|\n+\n|B||a_n-A|\\\\\n&<\n\\frac{\\epsilon}{2}\n+\n\\frac{|B|}{|B|+1}\\frac{\\epsilon}{2}\\\\\n&<\n\\epsilon.\n\\end{aligned}\n\\]\n\ub530\ub77c\uc11c \\(a_nb_n\\rightarrow AB\\)\uc774\ub2e4.<\/p>\n

    (5) \\(B\\ne0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(b_n\\rightarrow B\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|b_n-B| < \\frac{|B|}{2}.\n\\]\n\ub530\ub77c\uc11c\n\\[\n|b_n|\n\\ge\n|B|-|b_n-B|\n>
    \n\\frac{|B|}{2}>0.
    \n\\]
    \n\uadf8\ub7ec\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(b_n\\ne0\\)\uc774\ub2e4.<\/p>\n

    \ub610\ud55c
    \n\\[
    \n\\left|
    \n\\frac{1}{b_n}-\\frac{1}{B}
    \n\\right|
    \n=
    \n\\frac{|b_n-B|}{|b_n||B|}
    \n\\le
    \n\\frac{2}{|B|^2}|b_n-B|.
    \n\\]
    \n\ub530\ub77c\uc11c \\(b_n\\rightarrow B\\)\uc774\uba74
    \n\\[
    \n\\frac{1}{b_n}\\rightarrow\\frac{1}{B}.
    \n\\]
    \n\uc774 \uacb0\uacfc\uc640 (4)\ub97c \uc774\uc6a9\ud558\uba74
    \n\\[
    \n\\frac{a_n}{b_n}
    \n=
    \na_n\\frac{1}{b_n}
    \n\\rightarrow
    \nA\\frac{1}{B}
    \n=
    \n\\frac{A}{B}
    \n\\]
    \n\uc774\ub2e4.<\/p>\n

    (6) \\(m=1\\)\uc77c \ub54c\ub294 \uc790\uba85\ud558\ub2e4. \\(m>1\\)\uc77c \ub54c\uc5d0\ub294 (4)\ub97c \ubc18\ubcf5\ud574\uc11c \uc801\uc6a9\ud558\uba74
    \n\\[
    \n(a_n)^m
    \n=
    \n\\underbrace{a_n\\cdots a_n}_{m\\text{\uac1c}}
    \n\\rightarrow
    \n\\underbrace{A\\cdots A}_{m\\text{\uac1c}}
    \n=
    \nA^m
    \n\\]
    \n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

    (7) \uba3c\uc800 \\(A=0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc591\uc218 \\(\\epsilon\\)\uc774 \uc8fc\uc5b4\uc9c0\uba74 \\(a_n\\rightarrow0\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n0\\le a_n < \\epsilon^m.\n\\]\n\ub530\ub77c\uc11c\n\\[\n0\\le\\sqrt[m]{a_n}< \\epsilon\n\\]\n\uc774\ubbc0\ub85c\n\\[\n\\sqrt[m]{a_n}\\rightarrow0.\n\\]<\/p>\n

    \uc774\uc81c \\(A>0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \ub193\ub294\ub2e4.
    \n\\[
    \nx_n=\\sqrt[m]{a_n},
    \n\\qquad
    \nx=\\sqrt[m]{A}.
    \n\\]
    \n\uadf8\ub7ec\uba74 \\(x>0\\)\uc774\uace0
    \n\\[
    \na_n-A
    \n=
    \nx_n^m-x^m
    \n=
    \n(x_n-x)
    \n\\left(
    \nx_n^{m-1}+x_n^{m-2}x+\\cdots+x^{m-1}
    \n\\right).
    \n\\]
    \n\uad04\ud638 \uc548\uc758 \uac12\uc740 \\(x^{m-1}\\) \uc774\uc0c1\uc774\ubbc0\ub85c
    \n\\[
    \n|x_n-x|
    \n\\le
    \n\\frac{|a_n-A|}{x^{m-1}}.
    \n\\]
    \n\\(a_n\\rightarrow A\\)\uc774\ubbc0\ub85c \uc624\ub978\ucabd\uc740 \\(0\\)\uc73c\ub85c \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c
    \n\\[
    \n\\sqrt[m]{a_n}\\rightarrow\\sqrt[m]{A}.
    \n\\]
    \n<\/span><\/p>\n<\/div>\n