{"id":6707,"date":"2021-07-21T00:00:46","date_gmt":"2021-07-20T15:00:46","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6707"},"modified":"2026-09-27T20:20:30","modified_gmt":"2026-09-27T11:20:30","slug":"functions-of-several-variables","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/functions-of-several-variables\/","title":{"rendered":"\ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uadf9\ud55c"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p>\\[<br \/>\n\\newcommand{\\vecx}{{\\mathbf{x}}}<br \/>\n\\newcommand{\\vecc}{{\\mathbf{c}}}<br \/>\n\\newcommand{\\vecR}{{\\mathbb{R}}}<br \/>\n\\]\n<\/p><\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 4\uc7a5 4\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>\\(D\\)\uac00 \\(\\vecR^d\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(\\vecc\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \uc989 \uc138 \uc870\uac74<\/p>\n<ul>\n<li>\\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(\\vecc_n \\rightarrow \\vecc ,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\vecc_n \\ne \\vecc ,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\vecc_n \\in D\\)<\/li>\n<\/ul>\n<p>\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218\uc5f4 \\(\\left\\{ \\vecc_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<p>\uc774\uc81c \ud568\uc218 \\(f: D \\rightarrow \\mathbb{R}\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\vecx\\)\uac00 \\(\\vecx \\ne \\vecc\\)\uc774\uba74\uc11c \\(\\vecc\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc774 \ub2e4\uac00\uac08 \ub54c \ud568\uc22b\uac12 \\(f(\\vecx )\\)\uc774 \\(L\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc774 \ub2e4\uac00\uac00\uba74<\/p>\n<p class=\"aligncenter\">\u201c\\(\\vecc\\)\uc5d0\uc11c \\(f(\\vecx )\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\u201d<\/p>\n<p>\ub610\ub294<\/p>\n<p class=\"aligncenter\">\u201c\\(\\vecx\\)\uac00 \\(\\vecc\\)\uc5d0 \ub2e4\uac00\uac08 \ub54c \\(f(\\vecx )\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\u201d<\/p>\n<p>\ub610\ub294<\/p>\n<p class=\"aligncenter\">\u201c\\(\\vecx\\)\uac00 \\(\\vecc\\)\uc5d0 \ub2e4\uac00\uac08 \ub54c \\(f(\\vecx )\\)\uac00 \\(L\\)\uc5d0 \ub2e4\uac00\uac04\ub2e4\u201d<\/p>\n<p>\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4. \uc774\ub54c \\(L\\)\uc744 \u2018\\(\\vecc\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uadf9\ud55c<\/span>\u2019\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uac83\uc744 \uae30\ud638\ub85c<br \/>\n\\[<br \/>\n\\lim_{\\vecx\\rightarrow\\vecc} f(\\vecx ) = L<br \/>\n\\]<br \/>\n\ub610\ub294<br \/>\n\\[<br \/>\nf(\\vecx ) \\rightarrow L \\quad \\text{as} \\quad \\vecx \\rightarrow \\vecc<br \/>\n\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ubcc0\uc218\uac00 \ud558\ub098\uc778 \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uad6c\ud560 \ub54c \uc0ac\uc6a9\ud588\ub358 \ubc95\uce59\ub4e4\uc744 \ubcc0\uc218\uac00 \uc5ec\ub7ec \uac1c\uc778 \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uad6c\ud560 \ub54c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub2e4\uc74c \uc138 \ubcf4\uae30\ub97c \ud1b5\ud574 \uc774 \uc0ac\uc2e4\uc744 \ud655\uc778\ud558\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.4.1.<\/span><br \/>\n\\[<br \/>\n\\lim_{(x,\\,y)\\rightarrow (0,\\,1)} \\frac{x-xy+3}{x^2 y + 5xy &#8211; y^3}<br \/>\n=<br \/>\n\\frac{0-0\\times 1 + 3}{0^2 \\times 1 + 5 \\times 0 \\times 1 &#8211; 1^3}<br \/>\n=<br \/>\n-3.<br \/>\n\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.4.2.<\/span><br \/>\n\\[<br \/>\n\\lim_{(x,\\,y)\\rightarrow (3,\\,-4)} \\sqrt{x^2 + y^2}<br \/>\n=<br \/>\n\\sqrt{3^2 + (-4)^2}<br \/>\n=<br \/>\n\\sqrt{25}<br \/>\n=<br \/>\n5.<br \/>\n\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.4.3.<\/span><br \/>\n\uc8fc\uc5b4\uc9c4 \uc2dd\uc758 \uc815\uc758\uc5ed \uc548\uc5d0\uc11c\ub294 \\(x\\ge0,\\) \\(y\\ge0\\)\uc774\uace0 \\(x\\ne y\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n\\lim_{(x,\\,y)\\rightarrow (0,\\,0)} \\frac{x^2 -xy}{\\sqrt{x} &#8211; \\sqrt{y}}<br \/>\n&#038;= \\lim_{(x,\\,y)\\rightarrow (0,\\,0)} \\frac{(x^2 &#8211; xy)(\\sqrt{x} + \\sqrt{y})}{(\\sqrt{x} &#8211; \\sqrt{y})(\\sqrt{x} + \\sqrt{y} )} \\\\[4pt]<br \/>\n&#038;= \\lim_{(x,\\,y)\\rightarrow (0,\\,0)} \\frac{x(x-y)(\\sqrt{x} + \\sqrt{y})}{x-y} \\\\[4pt]<br \/>\n&#038;= \\lim_{(x,\\,y)\\rightarrow (0,\\,0)} x(\\sqrt{x} + \\sqrt{y}) \\\\[4pt]<br \/>\n&#038;= 0 \\times (\\sqrt{0} + \\sqrt{0} ) =0 .<br \/>\n\\end{align}<br \/>\n\\]\n<\/p>\n<\/div>\n<p>\ub2e4\uc74c \uc608\uc81c\ub294 \uc870\uc784 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uadf9\ud55c\uc744 \uad6c\ud558\ub294 \uc608\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 4.4.4.<\/span><br \/>\n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<br \/>\n\\[<br \/>\n\\lim_{(x,\\,y)\\rightarrow (0,\\,0)} \\frac{4xy^2}{x^2 +y^2}<br \/>\n\\]\n<\/p>\n<p class=\"margintop1\"><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\(x^2 + y^2 \\ne 0\\)\uc77c \ub54c<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n\\left\\lvert \\frac{4xy^2}{x^2 + y^2} \\right\\rvert<br \/>\n&#038;= \\frac{4\\lvert x \\rvert y^2}{x^2 + y^2} \\\\[4pt]<br \/>\n&#038;\\le \\frac{4\\lvert x \\rvert (x^2 + y^2 )}{x^2 + y^2} \\\\[4pt]<br \/>\n&#038;= 4\\lvert x \\rvert \\\\[4pt]<br \/>\n&#038;= 4\\sqrt{x^2} \\\\[4pt]<br \/>\n&#038;\\le 4\\sqrt{x^2 + y^2} \\\\[4pt]<br \/>\n&#038;= 4\\lVert (x,\\,y) \\rVert<br \/>\n\\end{align}<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n-4 \\lVert (x,\\,y) \\rVert<br \/>\n\\le<br \/>\n\\frac{4xy^2}{x^2 + y^2}<br \/>\n\\le<br \/>\n4\\lVert (x,\\,y) \\rVert<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\((x,\\,y)\\rightarrow (0,\\,0)\\)\uc77c \ub54c<br \/>\n\\[<br \/>\n\\frac{4xy^2}{x^2 + y^2} \\rightarrow 0<br \/>\n\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\ubcc0\uc218\uac00 \uc2e4\uc218\uc77c \ub54c\ub294 \ud55c \uc810\uc5d0 \ub2e4\uac00\uac00\ub294 \ubc29\ud5a5\uc744 \uc67c\ucabd\uacfc \uc624\ub978\ucabd\uc758 \ub450 \uac00\uc9c0\ub85c \ub098\ub204\uc5b4 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \ubcc0\uc218\uac00 \ub450 \uac1c \uc774\uc0c1\uc778 \uacbd\uc6b0, \uc989 \ud568\uc218\uc758 \uc815\uc758\uc5ed\uc774 \\(\\vecR^d\\)\uc774\uace0 \\(d \\ge 2\\)\uc778 \uacbd\uc6b0, \ubcc0\uc218\uac00 \ud55c \uc810\uc5d0 \ub2e4\uac00\uac00\ub294 \uacbd\ub85c\uac00 \uc5ec\ub7ec \uac00\uc9c0\uc774\ub2e4. \uc774 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf9\ud55c\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \ub2e4\uc74c \ub450 \uc608\uc81c\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 4.4.5.<\/span><br \/>\n\\(\\displaystyle f(x,\\,y) = \\frac{y}{x}\\)\uc77c \ub54c \uadf9\ud55c \\(\\displaystyle \\lim_{(x,\\,y)\\rightarrow (0,\\,0)} f(x,\\,y)\\)\ub97c \uc870\uc0ac\ud558\uc2dc\uc624.<\/p>\n<p class=\"margintop1\"><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\(y=x\\)\uc774\uba74\uc11c \\(x\\rightarrow 0\\)\uc774\uba74,<br \/>\n\\[<br \/>\n(x,\\,y) = (x,\\,x) \\rightarrow (0,\\,0)<br \/>\n\\]<br \/>\n\uc774\uba74\uc11c<br \/>\n\\[<br \/>\nf(x,\\,y) = \\frac{y}{x} = \\frac{x}{x} \\rightarrow 1<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ubc18\uba74 \\(y=-x\\)\uc774\uba74\uc11c \\(x\\rightarrow 0\\)\uc774\uba74,<br \/>\n\\[<br \/>\n(x,\\,y) = (x,\\,-x) \\rightarrow (0,\\,0)<br \/>\n\\]<br \/>\n\uc774\uba74\uc11c<br \/>\n\\[<br \/>\nf(x,\\,y) = \\frac{y}{x} = &#8211; \\frac{x}{x} \\rightarrow -1<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uc989 \\((x,\\,y)\\rightarrow (0,\\,0)\\)\uc77c \ub54c \ud568\uc22b\uac12 \\(f(x,\\,y)\\)\uc740 \ud558\ub098\uc758 \uac12\uc5d0 \ub2e4\uac00\uac00\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\lim_{(x,\\,y)\\rightarrow(0,\\,0)}f(x,\\,y)<br \/>\n\\]<br \/>\n\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 4.4.6.<\/span><br \/>\n\\(\\displaystyle f(x,\\,y) = \\frac{2x^2 y}{x^4 + y^2}\\)\uc77c \ub54c \uadf9\ud55c \\(\\displaystyle \\lim_{(x,\\,y)\\rightarrow (0,\\,0)} f(x,\\,y)\\)\ub97c \uc870\uc0ac\ud558\uc2dc\uc624.<\/p>\n<p class=\"margintop1\"><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\(y=x^2\\)\uc774\uba74\uc11c \\(x\\rightarrow 0\\)\uc774\uba74, \\((x,\\,y)\\rightarrow (0,\\,0)\\)\uc774\uba74\uc11c<br \/>\n\\[<br \/>\nf(x,\\,y) = \\frac{2x^2 y}{x^4 + y^2} = \\frac{2x^4}{2x^4}=1 \\rightarrow 1<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ubc18\uba74 \\(y=2x^2\\)\uc774\uba74\uc11c \\(x\\rightarrow 0\\)\uc774\uba74, \\((x,\\,y)\\rightarrow (0,\\,0)\\)\uc774\uba74\uc11c<br \/>\n\\[<br \/>\nf(x,\\,y) = \\frac{2x^2 y}{x^4 + y^2} = \\frac{4x^4}{5x^4}=\\frac{4}{5} \\rightarrow \\frac{4}{5}<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uc989 \\((x,\\,y)\\rightarrow (0,\\,0)\\)\uc77c \ub54c \ud568\uc22b\uac12 \\(f(x,\\,y)\\)\uc740 \ud558\ub098\uc758 \uac12\uc5d0 \ub2e4\uac00\uac00\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\lim_{(x,\\,y)\\rightarrow(0,\\,0)}f(x,\\,y)<br \/>\n\\]<br \/>\n\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.\n<\/p>\n<\/div>\n<p>\ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uc5f0\uc18d\uc131\ub3c4 \uc77c\ubcc0\uc218\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc815\uc758\ub41c\ub2e4. \ub354\uc6b1\uc774 \ub2eb\ud788\uace0 \uc720\uacc4\uc778 \uc9d1\ud569\uc5d0\uc11c \uc5f0\uc18d\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\ub294 \uc720\uacc4\uc774\uba70 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\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=\"..\/theorems-on-continuity\">\uc5f0\uc18d\uc131\uacfc \uad00\ub828\ub41c \uc815\ub9ac<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/vector-valued-functions\">\ubca1\ud130\ud568\uc218\uc758 \uadf9\ud55c<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\\( \\newcommand{\\vecx}{{\\mathbf{x}}} \\newcommand{\\vecc}{{\\mathbf{c}}} \\newcommand{\\vecR}{{\\mathbb{R}}} \\) \uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 4\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \\(D\\)\uac00 \\(\\vecR^d\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(\\vecc\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \uc989 \uc138 \uc870\uac74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(\\vecc_n \\rightarrow \\vecc ,\\) \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\vecc_n \\ne \\vecc ,\\) \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\vecc_n \\in D\\) \ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218\uc5f4 \\(\\left\\{ \\vecc_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\uc81c \ud568\uc218 \\(f: D \\rightarrow \\mathbb{R}\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\vecx\\)\uac00&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":404,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-6707","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6707","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=6707"}],"version-history":[{"count":22,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6707\/revisions"}],"predecessor-version":[{"id":10151,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6707\/revisions\/10151"}],"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=6707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}