{"id":6693,"date":"2021-07-20T23:57:11","date_gmt":"2021-07-20T14:57:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6693"},"modified":"2022-03-06T19:51:22","modified_gmt":"2022-03-06T10:51:22","slug":"properties-of-limit-of-a-function","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/properties-of-limit-of-a-function\/","title":{"rendered":"\ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 3\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\uc774 \uc808\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubd05\ub2c8\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uad00\uacc4<\/h2>\n<p>3.1\uc808\uc5d0\uc11c \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uc815\uc758\ud558\uae30 \uc704\ud558\uc5ec \uc9d1\uc801\uc810\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uc600\ub2e4. \uadf8\ub7f0\ub370 \uc9d1\uc801\uc810\uc740 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \ubc00\uc811\ud55c \uc5f0\uad00\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 3.2.1. (\uc218\uc5f4 \ud310\uc815\ubc95, Sequential Test)<\/span><\/p>\n<p>\\(f:D \\rightarrow \\mathbb{R}\\)\uac00 \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\uba70, \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \\(x \\rightarrow c\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc138 \uc870\uac74<\/p>\n<ul>\n<li>\\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(x_n \\rightarrow c,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\in D ,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\ne c\\)<\/li>\n<\/ul>\n<p>\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \uc218\uc5f4 \\(\\left\\{x_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \uc218\uc5f4 \\(\\left\\{ f(x_n ) \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.2.1.<\/span><br \/>\n\\(\\chi_\\mathbb{Q} : \\mathbb{R} \\rightarrow \\mathbb{R}\\)\uac00 \ud2b9\uc131\ud568\uc218(characteristic function)\ub77c\uace0 \ud558\uc790. \uc989<br \/>\n\\[\\chi_\\mathbb{Q} (x) =<br \/>\n\\begin{cases}<br \/>\n1 \\quad &#038; \\text{if} \\,\\, x\\in\\mathbb{Q} ,\\\\<br \/>\n0 \\quad &#038; \\text{if} \\,\\, x\\notin\\mathbb{Q}<br \/>\n\\end{cases}\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(c\\in \\mathbb{R}\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(c\\)\uc5d0 \uc218\ub834\ud558\uace0 \uc5b4\ub290 \ud56d\ub3c4 \\(c\\)\uc640 \uc77c\uce58\ud558\uc9c0 \uc54a\ub294 \uc720\ub9ac\uc218\uc5f4 \\(\\left\\{ q_n \\right\\}\\)\uacfc \ubb34\ub9ac\uc218\uc5f4 \\(\\left\\{ r_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc218\uc5f4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c \ub450 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{gather}<br \/>\n\\lim_{x\\rightarrow c}\\chi_\\mathbb{Q} (x) = \\lim_{n\\rightarrow \\infty} \\chi_\\mathbb{Q} (q_n ) = \\lim_{n\\rightarrow \\infty} 1 = 1 , \\\\[6pt]<br \/>\n\\lim_{x\\rightarrow c}\\chi_\\mathbb{Q} (x) = \\lim_{n\\rightarrow \\infty} \\chi_\\mathbb{Q} (r_n ) = \\lim_{n\\rightarrow \\infty} 0 = 0.<br \/>\n\\end{gather}\\]<br \/>\n\ub450 \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130 \\(1 = L = 0\\)\uc744 \uc5bb\ub294\ub370, \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x)\\)\ub294 \uc5b4\ub5a0\ud55c \uac12\uc5d0\ub3c4 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.\n<\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span> \uc810\uc5d0\uc11c \ubc1c\uc0b0\ud558\ub294 \ud568\uc218\uc758 \uadf9\ud55c\ub3c4 \uc218\uc5f4 \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \\(f:D \\rightarrow \\mathbb{R}\\)\uac00 \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c \uac01\uac01\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x) \\rightarrow \\infty\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(c\\)\uc5d0 \uc218\ub834\ud558\uace0 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\in D,\\) \\(x_n \\ne c\\)\uc778 \uc784\uc758\uc758 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \uc218\uc5f4 \\(\\left\\{ f(x_n ) \\right\\}\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\ub294 \uac83\uc774\ub2e4.<\/li>\n<li>\\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x) \\rightarrow -\\infty\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(c\\)\uc5d0 \uc218\ub834\ud558\uace0 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\in D,\\) \\(x_n \\ne c\\)\uc778 \uc784\uc758\uc758 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \uc218\uc5f4 \\(\\left\\{ f(x_n ) \\right\\}\\)\uc774 \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\ub294 \uac83\uc774\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc0ac\uce59\uacc4\uc0b0\uacfc \uad00\ub828\ub41c \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8<\/h2>\n<p>\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \ud568\uc218\uc758 \uadf9\ud55c\ub3c4 \uc0ac\uce59\uacc4\uc0b0\uacfc \uad00\ub828\ub41c \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 3.2.2. (\ud568\uc218\uc758 \uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8)<\/span><\/p>\n<p>\\(f\\)\uc640 \\(g\\)\uac00 \uc815\uc758\uc5ed \\(D\\)\ub97c \uacf5\ud1b5\uc73c\ub85c \uac16\ub294 \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(L,\\) \\(M\\)\uc774 \uc2e4\uc218\uc774\uba70<br \/>\n\\[\\lim_{x\\rightarrow c} f(x) = L \\quad\\text{and}\\quad \\lim_{x\\rightarrow c}g(x)=M\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\displaystyle \\lim_{x\\rightarrow c} (k f(x)) = kL.\\) &nbsp;(\\(k\\)\ub294 \uc2e4\uc218\uc778 \uc0c1\uc218.)<\/li>\n<li>\\(\\displaystyle \\lim_{x\\rightarrow c}(f(x)+g(x)) = L+M.\\)<\/li>\n<li>\\(\\displaystyle \\lim_{x\\rightarrow c}(f(x)-g(x)) = L-M.\\)<\/li>\n<li>\\(\\displaystyle \\lim_{x\\rightarrow c}(f(x)g(x)) = LM.\\)<\/li>\n<li>\\(\\displaystyle \\lim_{x\\rightarrow c}\\frac{f(x)}{g(x)} = \\frac{L}{M}.\\) &nbsp;(\ub2e8, \\(M\\ne 0.\\))<\/li>\n<li>\\(\\displaystyle \\lim_{x\\rightarrow c}\\sqrt[m]{f(x)} = \\sqrt[m]{L}.\\) &nbsp;(\ub2e8, \\(L\\ge 0\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ge 0.\\))<\/li>\n<\/ol>\n<\/div>\n<p>\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8(\uc815\ub9ac 1.3.1)\uacfc \uc218\uc5f4 \ud310\uc815\ubc95(\uc815\ub9ac 3.2.1)\uc744 \uacb0\ud569\ud558\uba74 \uc815\ub9ac 3.2.2\ub97c \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc138 \uc870\uac74<\/p>\n<ul>\n<li>\\(n \\rightarrow \\infty\\)\uc77c \ub54c \\(x_n \\rightarrow c,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\in D,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\ne c\\)<\/li>\n<\/ul>\n<p>\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc218\uc5f4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lim _ {n\\rightarrow\\infty} f(x_n ) = L \\quad\\text{and}\\quad \\lim_{n\\rightarrow\\infty} g(x_n ) = M\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lim_{n\\rightarrow\\infty}(f(x_n) + g(x_n))<br \/>\n=<br \/>\n\\lim_{n\\rightarrow\\infty}f(x_n) + \\lim_{n\\rightarrow\\infty}g(x_n)<br \/>\n=<br \/>\nL+M\\]<br \/>\n\uc774\ub2e4. \uc55e\uc758 \uc138 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc784\uc758\uc758 \uc218\uc5f4 \\(\\left\\{x_n\\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \uc774 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c \uc218\uc5f4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lim_{x\\rightarrow c}(f(x)+g(x)) = L+M\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 [2]\ub97c \uc99d\uba85\ud558\uc600\ub2e4. \ub2e4\ub978 \ub4f1\uc2dd\uc758 \uc99d\uba85\ub3c4 \uc774\uc640 \uac19\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.2.2.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\[\\begin{align}<br \/>\n\\lim_{x\\rightarrow 2} (x^2 -2x +3 )<br \/>\n&#038;= \\lim_{x\\rightarrow 2} x^2 &#8211; \\lim_{x\\rightarrow 2} 2x + \\lim_{x\\rightarrow 2} 3 \\\\[5pt]<br \/>\n&#038;= \\left( \\lim_{x\\rightarrow 2} x \\right) \\left( \\lim_{x\\rightarrow 2} x \\right) &#8211; 2 \\times \\left( \\lim_{x\\rightarrow 2} x \\right) + \\lim_{x\\rightarrow 2} 3 \\\\[5pt]<br \/>\n&#038;= 2 \\times 2 &#8211; 2 \\times 2 + 3 = 3.<br \/>\n\\end{align}\\]<\/li>\n<li>\\[\\begin{align}<br \/>\n\\lim_{x\\rightarrow -3} \\frac{x^2 -x+1}{x+1}<br \/>\n&#038;= \\frac{\\displaystyle \\lim_{x\\rightarrow -3} (x^2 -x+1)}{\\displaystyle \\lim_{x\\rightarrow -3} (x+1)} \\\\[5pt]<br \/>\n&#038;= \\frac{(-3)^2 -(-3) +1}{(-3)+1} = &#8211; \\frac{13}{2}.<br \/>\n\\end{align}\\]<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.2.3.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\[\\begin{align}<br \/>\n\\lim_{x\\rightarrow 1} \\frac{\\sqrt{x} -1}{x-1}<br \/>\n&#038;= \\lim_{x\\rightarrow 1} \\frac{(\\sqrt{x} -1)(\\sqrt{x}+1)}{(x-1)(\\sqrt{x}+1)} \\\\[5pt]<br \/>\n&#038;= \\lim_{x\\rightarrow 1} \\frac{x-1}{(x-1)(\\sqrt{x}+1)} \\\\[5pt]<br \/>\n&#038;= \\lim_{x\\rightarrow 1} \\frac{1}{\\sqrt{x}+1} = \\frac{1}{2}.<br \/>\n\\end{align}\\]<\/li>\n<li>\\[\\begin{align}<br \/>\n\\lim_{h\\rightarrow 0} \\frac{\\sqrt{2+h} &#8211; \\sqrt{2}}{h}<br \/>\n&#038;= \\lim_{h\\rightarrow 0} \\frac{(\\sqrt{2+h} &#8211; \\sqrt{2})(\\sqrt{2+h} + \\sqrt{2})}{h(\\sqrt{2+h} + \\sqrt{2})} \\\\[5pt]<br \/>\n&#038;= \\lim_{h\\rightarrow 0} \\frac{2+h-2}{h(\\sqrt{2+h} + \\sqrt{2})} \\\\[5pt]<br \/>\n&#038;= \\lim_{h\\rightarrow 0} \\frac{1}{\\sqrt{2+h} + \\sqrt{2}} \\\\[5pt]<br \/>\n&#038;= \\lim_{h\\rightarrow 0} \\frac{1}{\\sqrt{2} + \\sqrt{2}} = \\frac{1}{2\\sqrt{2}}.<br \/>\n\\end{align}\\]<\/li>\n<\/ol>\n<\/div>\n<div class=\"assignment\">\n<p><span class=\"definition\">\uacfc\uc81c.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\(p\\)\uac00 \ub2e4\ud56d\ud568\uc218\uc774\uace0 \\(c\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c<br \/>\n\\[\\lim_{x\\rightarrow c}p(x) = p(c)\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(p\\)\uc640 \\(q\\)\uac00 \ub2e4\ud56d\ud568\uc218\uc774\uace0 \\(c\\)\uac00 \uc2e4\uc218\uc774\uba70 \\(q(c) \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c<br \/>\n\\[\\lim_{x\\rightarrow c} \\frac{p(x)}{q(x)} = \\frac{p(c)}{q(c)}\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 3.2.4.<\/span><br \/>\n\ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub3c4\ub85d \ud558\ub294 \uc2e4\uc218 \\(a,\\) \\(b\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624.<br \/>\n\\[\\lim_{x\\rightarrow 2} \\frac{2x^2 +ax+b}{x-2} = 3.\\]<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span> \\(x\\rightarrow 2\\)\uc77c \ub54c \ubd84\ubaa8\uac00 \\(0\\)\uc5d0 \uc218\ub834\ud558\ub294\ub370 \ubd84\uc218\uc2dd\uc774 \ubc1c\uc0b0\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \ubd84\uc790 \ub610\ud55c \\(0\\)\uc5d0 \uc218\ub834\ud574\uc57c \ud55c\ub2e4. \uc989<br \/>\n\\[\\lim_{x\\rightarrow 2} (2x^2 +ax+b)=0\\]<br \/>\n\uc774\ub2e4. \uc774 \uc2dd\uc73c\ub85c\ubd80\ud130 \\(8+2a+b=0\\) \uc989 \\(b = -2a-8\\)\uc744 \uc5bb\ub294\ub2e4. \uc6d0\ub798\uc758 \uadf9\ud55c\uc5d0\uc11c \\(b\\)\ub97c \\(-2a-8\\)\ub85c \ubc14\uafb8\uba74 \ub2e4\uc74c \uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\lim_{x\\rightarrow 2}\\frac{2x^2 +ax-2a-8}{x-2} = 3.\\]<br \/>\n\uc774 \ub4f1\uc2dd\uc5d0\uc11c \uc88c\ubcc0\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\lim_{x\\rightarrow 2}\\frac{(2x+a+4)(x-2)}{x-2} = \\lim_{x\\rightarrow 2}(2x+a+4) = a+8.\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(a=-5,\\) \\(b=2\\)\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc21c\uc11c\uad00\uacc4\uc640 \uad00\ub828\ub41c \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8<\/h2>\n<p>\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \ud568\uc218\uc758 \uadf9\ud55c\ub3c4 \uc21c\uc11c\uad00\uacc4\uc640 \uad00\ub828\ub41c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 3.2.3. (\uc21c\uc11c\ubcf4\uc874 \uc131\uc9c8)<\/span><\/p>\n<p>\\(f\\)\uc640 \\(g\\)\uac00 \uacf5\ud1b5\uc815\uc758\uc5ed \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in (I\\cap D)\\setminus\\left\\{ c \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\le g(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(g(x)\\)\uac00 \uac01\uac01 \\(L,\\) \\(M\\)\uc5d0 \uc218\ub834\ud558\uba74 \\(L\\le M\\)\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x)\\)\uac00 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uba74 \\(g(x)\\)\ub3c4 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(x\\rightarrow c\\)\uc77c \ub54c \\(g(x)\\)\uac00 \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uba74 \\(f(x)\\)\ub3c4 \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<div style=\"margin-top: 1.5em;\"><img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img01.png\" alt=\"\" width=\"301\" height=\"225\" class=\"aligncenter size-full wp-image-7252\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img01.png 1504w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img01-300x225.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img01-1024x767.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img01-768x575.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img01-1170x877.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img01-585x438.png 585w\" sizes=\"(max-width: 301px) 100vw, 301px\" \/><\/div>\n<\/div>\n<p>\uc138 \uac1c\uc758 \ud568\uc218\ub97c \ube44\uad50\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 3.2.4. (\uc870\uc784 \uc815\ub9ac)<\/span><\/p>\n<p>\\(f,\\) \\(g,\\) \\(h\\)\uac00 \uacf5\ud1b5\uc815\uc758\uc5ed \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in (I\\cap D)\\setminus\\left\\{ c \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\le g(x) \\le h(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x)\\)\uc640 \\(h(x)\\)\uac00 \uac19\uc740 \uac12 \\(L\\)\uc5d0 \uc218\ub834\ud558\uba74, \\(x\\rightarrow c\\)\uc77c \ub54c \\(g(x)\\)\ub3c4 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<div style=\"margin-top: 1.5em;\"><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02.png\" alt=\"\" width=\"318\" height=\"225\" class=\"aligncenter size-full wp-image-7253\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02.png 1588w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02-300x213.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02-1024x727.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02-768x545.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02-1536x1090.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02-1170x830.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img02-585x415.png 585w\" sizes=\"(max-width: 318px) 100vw, 318px\" \/><\/div>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.2.5.<\/span><br \/>\n\ud568\uc218 \\(f:\\mathbb{R}^{\\times} \\rightarrow \\mathbb{R}\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[f(x) = x\\sin \\frac{1}{x}.\\]<br \/>\n(\uc5ec\uae30\uc11c \\(\\mathbb{R}^\\times = \\mathbb{R} \\setminus\\left\\{ 0 \\right\\}\\)\uc774\ub2e4.)<br \/>\n\\[g(x) = \\lvert x \\rvert\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(x\\in\\mathbb{R}^\\times\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[-g(x) \\le f(x) \\le g(x)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\lim_{x\\rightarrow 0} (-g(x)) = \\lim_{x\\rightarrow 0} g(x) =0\\]<br \/>\n\uc774\ubbc0\ub85c, \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lim_{x\\rightarrow 0} g(x)=0\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p class=\"marginbottom2\"><span class=\"example\">\uc608\uc81c 3.2.6.<\/span><br \/>\n\ub2e4\uc74c \uadf9\ud55c\uc744 \uc870\uc0ac\ud558\uc2dc\uc624.<br \/>\n\\[\\lim_{x\\rightarrow 0}\\frac{\\sin x}{x}.\\]<br \/>\n(\uc5ec\uae30\uc11c \u2018\uadf9\ud55c\uc744 \uc870\uc0ac\ud55c\ub2e4\u2019\ub77c\ub294 \ub9d0\uc740 \uadf9\ud55c\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uace0, \uc218\ub834\ud55c\ub2e4\uba74 \uadf9\ud55c\uac12\uc744 \uad6c\ud558\ub77c\ub294 \ub73b\uc774\ub2e4.)<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\(x\\)\uac00 \\(0\\)\uc5d0 \uac00\uae4c\uc6b8 \ub54c\ub9cc \uc0dd\uac01\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \uba3c\uc800 \\(0 < x < \\pi \/2\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\uc544\ub798 \uadf8\ub9bc\uc744 \ubcf4\uc790.<\/p>\n<div style=\"margin-top: 1.5em; margin-bottom: 1.5em;\"><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img03.png\" alt=\"\" width=\"190\" height=\"163\" class=\"aligncenter size-full wp-image-7254\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img03.png 1137w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img03-300x258.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img03-1024x881.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img03-768x661.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/ic_ch03_se02_img03-585x503.png 585w\" sizes=\"(max-width: 190px) 100vw, 190px\" \/><\/div>\n<ul>\n<li>\uc0bc\uac01\ud615 \\(\\mathrm{ABC}\\)\uc758 \ub113\uc774\ub294 \\(\\frac{1}{2}\\sin x\\)\uc774\ub2e4.<\/li>\n<li>\ubd80\ucc44\uaf34 \\(\\mathrm{ABC}\\)\uc758 \ub113\uc774\ub294 \\(\\frac{1}{2} x\\)\uc774\ub2e4.<\/li>\n<li>\uc0bc\uac01\ud615 \\(\\mathrm{ABD}\\)\uc758 \ub113\uc774\ub294 \\(\\frac{1}{2}\\tan x\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uadf8\ub9bc\uc5d0\uc11c \ub113\uc774\ub97c \ube44\uad50\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\frac{1}{2} \\tan x \\ge \\frac{1}{2} x \\ge \\frac{1}{2} \\sin x .\\]<br \/>\n\uac01 \uc2dd\uc744 \\(\\frac{1}{2} \\sin x\\)\ub85c \ub098\ub204\uace0 \uc5ed\uc218\ub97c \ucde8\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\cos x \\le \\frac{\\sin x}{x} \\le 1.\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\cos x\\)\uc640 \\(\\frac{\\sin x}{x}\\)\uac00 \ubaa8\ub450 \uc6b0\ud568\uc218\uc774\ubbc0\ub85c, \uc704 \ubd80\ub4f1\uc2dd\uc740 \\( &#8211; \\frac{\\pi}{2} < x < 0\\)\uc77c \ub54c\ub3c4 \uc131\ub9bd\ud55c\ub2e4. \ub354\uc6b1\uc774\n\\[\\lim_{x\\rightarrow 0}\\cos x = 1 \\quad\\text{and}\\quad \\lim_{x\\rightarrow 0} 1 = 1\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec\n\\[\\lim_{x\\rightarrow 0} \\frac{\\sin x}{x} = 1\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p class=\"marginbottom2\"><span class=\"example\">\uc608\uc81c 3.2.7.<\/span><br \/>\n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<br \/>\n\\[\\lim_{x\\rightarrow 0} \\frac{1-\\cos x}{x} .\\]<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\uadf9\ud55c\uc744 \uad6c\ud560 \ubd84\uc218\uc2dd\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\frac{1-\\cos x}{x}<br \/>\n&#038;= \\frac{(1-\\cos x)(1+ \\cos x)}{x(1+\\cos x)} \\\\[5pt]<br \/>\n&#038;= \\frac{1-\\cos ^2 x}{x(1+\\cos x)} \\\\[5pt]<br \/>\n&#038;= \\frac{\\sin^2 x}{x(1+ \\cos x )}.<br \/>\n\\end{align}\\]<br \/>\n\uc774 \uacb0\uacfc\ub97c \uc774\uc6a9\ud558\uc5ec \ubb38\uc81c\uc758 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{x\\rightarrow 0}\\frac{1-\\cos x }{x}<br \/>\n&#038;= \\lim_{x\\rightarrow 0} \\frac{\\sin^2 x}{x(1+\\cos x)} \\\\[5pt]<br \/>\n&#038;= \\lim_{x\\rightarrow 0} \\left\\{ \\frac{\\sin x}{x} \\times \\frac{\\sin x}{1+\\cos x} \\right\\} \\\\[5pt]<br \/>\n&#038;= 1\\times \\frac{0}{2} = 0.<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n\n\n\n<p>.<\/p>\n\n\n\n\n\n<p>.<\/p>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/limit-of-a-function-at-a-point\">\uc810\uc5d0\uc11c \ud568\uc218\uc758 \uadf9\ud55c<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/one-sided-limits\">\uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 3\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uc774 \uc808\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubd05\ub2c8\ub2e4. \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uad00\uacc4 3.1\uc808\uc5d0\uc11c \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uc815\uc758\ud558\uae30 \uc704\ud558\uc5ec \uc9d1\uc801\uc810\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uc600\ub2e4. \uadf8\ub7f0\ub370 \uc9d1\uc801\uc810\uc740 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \ubc00\uc811\ud55c \uc5f0\uad00\uc744 \uac00\uc9c4\ub2e4. \uc815\ub9ac 3.2.1. (\uc218\uc5f4 \ud310\uc815\ubc95, Sequential Test) \\(f:D \\rightarrow \\mathbb{R}\\)\uac00 \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\uba70, \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \\(x \\rightarrow c\\)\uc77c \ub54c&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":302,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6693","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6693","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"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=6693"}],"version-history":[{"count":37,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6693\/revisions"}],"predecessor-version":[{"id":8385,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6693\/revisions\/8385"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6693"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}