{"id":4449,"date":"2020-05-03T12:58:11","date_gmt":"2020-05-03T03:58:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4449"},"modified":"2020-09-14T13:08:36","modified_gmt":"2020-09-14T04:08:36","slug":"calculus-exercises-on-the-limits-of-a-multivariable-functions","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-exercises-on-the-limits-of-a-multivariable-functions\/","title":{"rendered":"\ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uadf9\ud55c \uc99d\uba85 \uc5f0\uc2b5\ubb38\uc81c"},"content":{"rendered":"<style type=\"text\/css\">\ndiv.problemquestion {\n\tmargin-bottom: 3em;\n}\ndiv.problemquestion p:last-child {\n\tmargin-bottom: 0em;\n}\n<\/style>\n<div class=\"problemquestion\">\n<p>\ubbf8\uc801\ubd84\ud559 II \uacfc\uc81c\uc785\ub2c8\ub2e4. \ub2e4\uc74c \ubb38\uc81c \uc911 \ud558\ub098 \uc774\uc0c1\uc744 \ud480\uc5b4\uc11c \uc81c\ucd9c\ud558\uc138\uc694. \ubb38\uc81c\uc5d0\uc11c \\(f\\)\ub294 \ud56d\uc0c1 \uc815\uc758\uc5ed\uc774 \\(\\mathbb{R}^2\\)\uc774\uace0 \uacf5\uc5ed\uc774 \\(\\mathbb{R}\\)\uc778 \ud568\uc218\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \ub610\ud55c \uc99d\uba85\uc740 \ubaa8\ub450 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc758 \uc5c4\ubc00\ud55c \uc815\uc758(\\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95)\ub97c \uc0ac\uc6a9\ud574\uc57c \ud569\ub2c8\ub2e4. (\ubb38\uc81c\uc5d0\uc11c \u2018\ucc38\uace0\u2019\ub294 \ud78c\ud2b8\uac00 \uc544\ub2d9\ub2c8\ub2e4.)<\/p>\n<\/div>\n<div class=\"problemquestion\">\n<p><span class=\"definition\">\ubb38\uc81c 1.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) = \\sin x + \\sin y\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \ubaa8\ub4e0 \uc810 \\((x,\\,y)\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problemquestion\">\n<p><span class=\"definition\">\ubb38\uc81c 2.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\n1 \\quad &#038; \\text{if} \\,\\, y=x^2 \\text{ and } x > 0 \\\\[6pt]<br \/>\n0 \\quad &#038; \\text{otherwise}<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. <br \/>[\ucc38\uace0: \uc774 \ud568\uc218\ub294 \\((0,\\,0)\\)\uc5d0\uc11c \ubaa8\ub4e0 \ubc29\ud5a5\uc73c\ub85c\uc758 \ubc29\ud5a5\ubbf8\ubd84\uacc4\uc218\uac00 \\(0\\)\uc774\ub2e4.]<\/p>\n<\/div>\n<div class=\"problemquestion\">\n<p><span class=\"definition\">\ubb38\uc81c 3.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\n\\frac{x^4 + y^4}{x^2 + y^2} \\quad &#038; \\text{if} \\,\\, (x,\\,y) \\ne (0,\\,0) \\\\[6pt]<br \/>\n0 \\quad &#038; \\text{otherwise}<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problemquestion\">\n<p><span class=\"definition\">\ubb38\uc81c 4.<\/span> \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<br \/>\n\\[f(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\nx^2 + y^2 \\quad &#038; \\text{if} \\,\\, (x,\\,y) \\in \\mathbb{Q} \\times \\mathbb{Q} \\\\[6pt]<br \/>\n0 \\quad &#038; \\text{otherwise}<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub54c, \\(f\\)\uac00 \\((0,\\,0)\\)\uc5d0\uc11c\ub9cc \uc5f0\uc18d\uc774\uace0 \ub2e4\ub978 \uc810\uc5d0\uc11c\ub294 \ubd88\uc5f0\uc18d\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<br \/>\n<br \/>[\ucc38\uace0: \uc774 \ud568\uc218\ub294 \\((0,\\,0)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4.]<\/p>\n<\/div>\n<div class=\"box\">\n<p class=\"aligncenter\"><a href=\"\/blog\/articles\/calculus-exercises-on-the-limits-of-a-multivariable-functions-solution\/\">\ud480\uc774 \ubcf4\uae30<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\ubbf8\uc801\ubd84\ud559 II \uacfc\uc81c\uc785\ub2c8\ub2e4. \ub2e4\uc74c \ubb38\uc81c \uc911 \ud558\ub098 \uc774\uc0c1\uc744 \ud480\uc5b4\uc11c \uc81c\ucd9c\ud558\uc138\uc694. \ubb38\uc81c\uc5d0\uc11c \\(f\\)\ub294 \ud56d\uc0c1 \uc815\uc758\uc5ed\uc774 \\(\\mathbb{R}^2\\)\uc774\uace0 \uacf5\uc5ed\uc774 \\(\\mathbb{R}\\)\uc778 \ud568\uc218\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \ub610\ud55c \uc99d\uba85\uc740 \ubaa8\ub450 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc758 \uc5c4\ubc00\ud55c \uc815\uc758(\\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95)\ub97c \uc0ac\uc6a9\ud574\uc57c \ud569\ub2c8\ub2e4. (\ubb38\uc81c\uc5d0\uc11c \u2018\ucc38\uace0\u2019\ub294 \ud78c\ud2b8\uac00 \uc544\ub2d9\ub2c8\ub2e4.) \ubb38\uc81c 1. \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4. \\(f(x,\\,y) = \\sin x + \\sin y\\) \uc774\ub54c, \\(f\\)\uac00 \ubaa8\ub4e0 \uc810 \\((x,\\,y)\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ubb38\uc81c 2. \ud568\uc218 \\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4&hellip;<\/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":[],"class_list":["post-4449","post","type-post","status-publish","format-standard","hentry","category-calculus-ap"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4449","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=4449"}],"version-history":[{"count":23,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4449\/revisions"}],"predecessor-version":[{"id":5298,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4449\/revisions\/5298"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4449"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4449"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4449"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}