test_math_compare.jsonl•113 kB
{"track_id": "0631a2db-134c-433f-b10f-59cb56f442a6", "url": "https://socratic.org/questions/do-f-x-6-10x-2-and-g-x-8-x-2-2-share-any-tangent-lines#265435", "html": "\n\n\n<!DOCTYPE html>\n<html \n itemscope itemtype=\"http://schema.org/QAPage\"\n>\n <head>\n <!-- custom metatags -->\n \n \n\n\n <meta charset=\"utf-8\">\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=edge\">\n\n<!-- about the page -->\n<title>Do f(x) = 6 – 10x^2 and g(x) = 8 – (x – 2)^2 share any tangent lines? | Socratic</title>\n<meta name=\"description\"\n content=\"Yes, they do. The lines are (approximately) y=7.741x+7.498 and y=1.148x+6.033 Function f f(x) = 6-10x^2 so f'(x) = -20x At a point (a,f(a)), the equation of the tangent line is y=f(a)+f'(a)(x-a) = (6-10a^2)+(-20a)(x-a) which can be simplified to y= - 20ax+10a^2+6 Function g g(x) = 4+4x-x^2 so g'(x) = 4-2x At a point (b,g(b)), the equation of the tangent line is y=g(b)+g'(b)(x-b) = (4+4b-b^2)+(4-2b)(x-b) which can be simplified to y = (4-2b)x +b^2+4 The lines coincide when their slopes are the same and their y intercepts are the same. -20a = 4-2b which implies b=10a+2 10a^2+6=b^2+4 = (10a+2)^2+4 This leads to 90a^2+40a+2=0 The solutions are a=(-10+-sqrt(55))/45 and b=10a+2 Using the approximations a ~~ -0.0374178 and a ~~ -0.387027 we get corresponding values of b ~~ 1.87027 and b ~~ -1.87027 Using these in y= - 20ax+10a^2+6 or y = (4-2b)x +b^2+4 gets us the lines y=1.148x+6.033 and y=7.741x+7.498\"/>\n\n<!-- canonical url -->\n\n<link href=\"https://socratic.org/questions/do-f-x-6-10x-2-and-g-x-8-x-2-2-share-any-tangent-lines\" rel=\"canonical\" />\n\n\n<!-- favicon -->\n\n <link rel=\"icon\" type=\"image/png\"\n href=\"/static/images/favicon32-qa.png\" />\n\n<!-- search engine site verification codes -->\n<meta name=\"google-site-verification\" content=\"KPpRSeYqJagzMJIt2YcPHKaz9bemdrB9ZPf0y53CC2E\" />\n<meta name=\"msvalidate.01\" content=\"0451EDE6FAD80488BAFF9B731C685FFC\" />\n\n\n \n\n\n\n\n<!-- Facebook Open Graph meta tags - these are also used by Twitterbot -->\n<meta property=\"fb:app_id\" content=\"\" />\n<meta property=\"og:site_name\" content=\"Socratic.org\">\n<meta property=\"og:type\" content=\"website\">\n<meta property=\"og:title\" itemprop=\"name\" content=\"Do f(x) = 6 – 10x^2 and g(x) = 8 – (x – 2)^2 share any tangent lines? | Socratic\">\n<meta property=\"og:description\" content=\"Yes, they do. The lines are (approximately) y=7.741x+7.498 and y=1.148x+6.033 Function f f(x) = 6-10x^2 so f'(x) = -20x At a point (a,f(a)), the equation of the tangent line is y=f(a)+f'(a)(x-a) = (6-10a^2)+(-20a)(x-a) which can be simplified to y= - 20ax+10a^2+6 Function g g(x) = 4+4x-x^2 so g'(x) = 4-2x At a point (b,g(b)), the equation of the tangent line is y=g(b)+g'(b)(x-b) = (4+4b-b^2)+(4-2b)(x-b) which can be simplified to y = (4-2b)x +b^2+4 The lines coincide when their slopes are the same and their y intercepts are the same. -20a = 4-2b which implies b=10a+2 10a^2+6=b^2+4 = (10a+2)^2+4 This leads to 90a^2+40a+2=0 The solutions are a=(-10+-sqrt(55))/45 and b=10a+2 Using the approximations a ~~ -0.0374178 and a ~~ -0.387027 we get corresponding values of b ~~ 1.87027 and b ~~ -1.87027 Using these in y= - 20ax+10a^2+6 or y = (4-2b)x +b^2+4 gets us the lines y=1.148x+6.033 and y=7.741x+7.498\">\n<meta property=\"og:image\" itemprop=\"image primaryImageOfPage\" 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itemprop=\"name\">Do #f(x) = 6 – 10x^2# and #g(x) = 8 – (x – 2)^2 # share any tangent lines?</h1>\n \n\n <div class=\"clearfix\">\n \n\n<div class='breadcrumbs weak-text'>\n \n <span itemscope itemtype=\"http://data-vocabulary.org/Breadcrumb\">\n <a href=\"/calculus\" itemprop=\"url\">\n <span class=\"breadcrumb\" itemprop=\"title\">Calculus</span>\n </a>\n </span>\n\n \n <span class='breadcrumb-gray'></span>\n \n <span itemscope itemtype=\"http://data-vocabulary.org/Breadcrumb\">\n <a href=\"/calculus#derivatives\" itemprop=\"url\">\n <span class=\"breadcrumb\" itemprop=\"title\">Derivatives</span>\n </a>\n </span>\n\n <span class='breadcrumb-gray'></span>\n \n <span itemscope itemtype=\"http://data-vocabulary.org/Breadcrumb\">\n <a href=\"/calculus/derivatives/tangent-line-to-a-curve\" itemprop=\"url\">\n <span class=\"breadcrumb\" itemprop=\"title\">Tangent Line to a Curve</span>\n </a>\n </span>\n\n \n</div>\n </div>\n\n \n\n <div id=\"main\" class='clearfix'>\n <div class='questionPageLeft floatLeft'>\n <h5 class=\"answersHeader hidesWhenAnswerEditing\">\n <span itemprop=\"answerCount\">2</span> Answers\n </h5>\n <div class=\"answers\">\n \n \n\n<div id=\"265411\" class=\"answer\" itemprop=\"suggestedAnswer\" itemscope itemtype=\"http://schema.org/Answer\">\n <div class=\"hidesWhenAnswerEditing\">\n <div class=\"answerTop clearfix\">\n \n \n \n \n \n <div class=\"contributorPic\">\n <a href=\"/users/jimh\" class=\"topContributorPic\" ><img src=\"https://profilepictures.socratic.org/V9qrDvNyQe6yk76DuTmQ_cropped%20Galileo_Poster-1.jpg\" title=\"\" alt=\"\"\n \n class=\"\"\n ></a>\n \n </div>\n \n <div class=\"answerInfo\">\n <div class=\"answerInfoTop clearfix\"><a href=\"/users/jimh\"\n \n class=\"topContributorLink\">\n Jim H\n </a></div>\n <div class=\"answerInfoBottom clearfix weak-text\">\n <span itemprop=\"dateCreated\" datetime=\"2016-05-13T16:57:24\" class=\"dateCreated\">\n May 13, 2016\n </span>\n </div>\n </div>\n </div>\n <div class=\"answerContainer clearfix\">\n <div class='answerText' itemprop=\"text\">\n \n <div class=\"answerSummary\">\n <div>\n <div class='markdown'><p>Yes, they do. The lines are (approximately) <mathjax>#y=7.741x+7.498#</mathjax> and <mathjax>#y=1.148x+6.033#</mathjax></p></div>\n </div>\n </div>\n \n \n <div class=\"answerDescription\">\n \n <h4 class=\"answerHeader\">Explanation:</h4>\n \n <div>\n <div class='markdown'><p><strong>Function <mathjax>#f#</mathjax></strong></p>\n<p><mathjax>#f(x) = 6-10x^2#</mathjax> so <mathjax>#f'(x) = -20x#</mathjax></p>\n<p>At a point <mathjax>#(a,f(a))#</mathjax>, the equation of the tangent line is </p>\n<p><mathjax>#y=f(a)+f'(a)(x-a)#</mathjax></p>\n<blockquote>\n<p><mathjax># = (6-10a^2)+(-20a)(x-a)#</mathjax> which can be simplified to </p>\n</blockquote>\n<p><mathjax># y= - 20ax+10a^2+6#</mathjax></p>\n<p><strong>Function <mathjax>#g#</mathjax><br />\n</strong><br />\n<mathjax>#g(x) = 4+4x-x^2#</mathjax> so <mathjax>#g'(x) = 4-2x#</mathjax></p>\n<p>At a point <mathjax>#(b,g(b))#</mathjax>, the equation of the tangent line is </p>\n<p><mathjax>#y=g(b)+g'(b)(x-b)#</mathjax></p>\n<blockquote>\n<p><mathjax># = (4+4b-b^2)+(4-2b)(x-b)#</mathjax> which can be simplified to </p>\n</blockquote>\n<p><mathjax>#y = (4-2b)x +b^2+4#</mathjax></p>\n<p><strong>The lines coincide</strong> when their slopes are the same and their <mathjax>#y#</mathjax> intercepts are the same.</p>\n<p><mathjax>#-20a = 4-2b#</mathjax> which implies <mathjax>#b=10a+2#</mathjax></p>\n<p><mathjax>#10a^2+6=b^2+4 = (10a+2)^2+4#</mathjax></p>\n<p>This leads to </p>\n<p><mathjax>#90a^2+40a+2=0#</mathjax></p>\n<p>The solutions are <mathjax>#a=(-10+-sqrt(55))/45#</mathjax> and <mathjax>#b=10a+2#</mathjax></p>\n<p>Using the approximations <mathjax>#a ~~ -0.0374178#</mathjax> and <mathjax>#a ~~ -0.387027#</mathjax></p>\n<p>we get corresponding values of <mathjax>#b ~~ 1.87027#</mathjax> and <mathjax>#b ~~ -1.87027#</mathjax></p>\n<p>Using these in <mathjax># y= - 20ax+10a^2+6#</mathjax> or <mathjax>#y = (4-2b)x +b^2+4#</mathjax> gets us the lines</p>\n<p><mathjax>#y=1.148x+6.033#</mathjax> and </p>\n<p><mathjax>#y=7.741x+7.498#</mathjax></p></div>\n </div>\n </div>\n \n </div>\n </div>\n </div>\n <div>\n <a itemprop=\"url\" href=\"https://socratic.org/answers/265411\">Answer link</a>\n </div>\n</div>\n \n \n\n<div id=\"265435\" class=\"answer\" itemprop=\"suggestedAnswer\" itemscope itemtype=\"http://schema.org/Answer\">\n <div class=\"hidesWhenAnswerEditing\">\n <div class=\"answerTop clearfix\">\n \n \n \n \n \n <div class=\"contributorPic\">\n <a href=\"/users/cesareo-r\" class=\"topContributorPic\" ><img src=\"https://profilepictures.socratic.org/oGfNPK9RStCHtrapdftB_yo.jpeg\" title=\"\" alt=\"\"\n \n class=\"\"\n ></a>\n \n </div>\n \n <div class=\"answerInfo\">\n <div class=\"answerInfoTop clearfix\"><a href=\"/users/cesareo-r\"\n \n class=\"topContributorLink\">\n Cesareo R.\n </a></div>\n <div class=\"answerInfoBottom clearfix weak-text\">\n <span itemprop=\"dateCreated\" datetime=\"2016-05-13T18:10:05\" class=\"dateCreated\">\n May 13, 2016\n </span>\n </div>\n </div>\n </div>\n <div class=\"answerContainer clearfix\">\n <div class='answerText' itemprop=\"text\">\n \n <div class=\"answerSummary\">\n <div>\n <div class='markdown'><p>There are two solutions as detailed below</p></div>\n </div>\n </div>\n \n \n <div class=\"answerDescription\">\n \n <h4 class=\"answerHeader\">Explanation:</h4>\n \n <div>\n <div class='markdown'><p>The tangent lines to <mathjax>#f#</mathjax> and <mathjax>#g#</mathjax> are given by<br />\n<mathjax>#t_f->(y-f(a))=f_x(a)(x-a)#</mathjax><br />\n<mathjax>#t_g->(y-g(b))=g_x(b)(x-b)#</mathjax> where <mathjax>#f_x,g_x#</mathjax> indicates derivatives with respect to <mathjax>#x#</mathjax>.<br />\nthe equations read<br />\n<mathjax>#t_f->y-6 + 10 a^2 = -20 a (x-a)#</mathjax><br />\n<mathjax>#t_g->y-8 + (b-2)^2 = -2 (b-2) (x-b)#</mathjax><br />\nEquating the <mathjax>#y#</mathjax> values<br />\n<mathjax>#2 (3 + 5 a^2 - 10 a x)=4 + b^2 + (4 - 2 b) x#</mathjax><br />\nand imposing the equality for all <mathjax>#x#</mathjax> we obtain the conditions<br />\n<mathjax>#2 + 10 a^2 - b^2=0,20 a + 4-2 b=0#</mathjax><br />\nSolving for <mathjax>#a,b#</mathjax> we obtain two solutions:<br />\n<mathjax>#{a -> 1/45 (-10 - sqrt[55]), b -> 2/9 (-1 - sqrt[55])}#</mathjax> and<br />\n<mathjax>#{a -> 1/45 (-10 + sqrt[55]), b -> 2/9 (-1 + sqrt[55])}#</mathjax></p></div>\n </div>\n </div>\n \n </div>\n </div>\n </div>\n <div>\n <a itemprop=\"url\" href=\"https://socratic.org/answers/265435\">Answer link</a>\n </div>\n</div>\n \n </div>\n </div>\n\n <div class='questionPageRight floatRight hidesWhenAnswerEditing'>\n <div class='questionPageRightMain'>\n \n <div class=\"relatedQuestions bordered\">\n <h5>Related questions</h5>\n <ul id=\"relatedQuestions\">\n \n <li class=\"relatedQuestionItem\">\n <div class=\"questionLink\">\n <a href=\"/questions/how-do-you-find-the-equation-of-a-tangent-line-to-a-curve\" class=\"relatedQuestionLink\">\n How do you find the equation of a tangent line to a curve?\n </a>\n </div>\n </li>\n 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'.markdown').replaceWith(imageAttributionReplacer);\n });\n </script>\n\n\n </body>\n</html>", "webkit_extract_md": "# Do #f(x) = 6 – 10x^2# and #g(x) = 8 – (x – 2)^2 # share any tangent lines?\n\nYes, they do. The lines are (approximately)\n\n\\#y=7.741x+7.498\\#\n\nand\n\n\\#y=1.148x+6.033\\#\n\n#### Explanation:\n\nFunction\n\n\\#f\\#\n\n\\#f(x) = 6-10x^2\\#\n\nso\n\n\\#f'(x) = -20x\\#\n\nAt a point\n\n\\#(a,f(a))\\#\n\n, the equation of the tangent line is\n\n\\#y=f(a)+f'(a)(x-a)\\#\n\n\\# = (6-10a^2)+(-20a)(x-a)\\#\n\nwhich can be simplified to\n\n\\# y= - 20ax+10a^2+6\\#\n\nFunction\n\n\\#g\\#\n\n\\#g(x) = 4+4x-x^2\\#\n\nso\n\n\\#g'(x) = 4-2x\\#\n\nAt a point\n\n\\#(b,g(b))\\#\n\n, the equation of the tangent line is\n\n\\#y=g(b)+g'(b)(x-b)\\#\n\n\\# = (4+4b-b^2)+(4-2b)(x-b)\\#\n\nwhich can be simplified to\n\n\\#y = (4-2b)x +b^2+4\\#\n\nThe lines coincide when their slopes are the same and their\n\n\\#y\\#\n\nintercepts are the same.\n\n\\#-20a = 4-2b\\#\n\nwhich implies\n\n\\#b=10a+2\\#\n\n\\#10a^2+6=b^2+4 = (10a+2)^2+4\\#\n\nThis leads to\n\n\\#90a^2+40a+2=0\\#\n\nThe solutions are\n\n\\#a=(-10+-sqrt(55))/45\\#\n\nand\n\n\\#b=10a+2\\#\n\nUsing the approximations\n\n\\#a ~~ -0.0374178\\#\n\nand\n\n\\#a ~~ -0.387027\\#\n\nwe get corresponding values of\n\n\\#b ~~ 1.87027\\#\n\nand\n\n\\#b ~~ -1.87027\\#\n\nUsing these in\n\n\\# y= - 20ax+10a^2+6\\#\n\nor\n\n\\#y = (4-2b)x +b^2+4\\#\n\ngets us the lines\n\n\\#y=1.148x+6.033\\#\n\nand\n\n\\#y=7.741x+7.498\\#\n\nThere are two solutions as detailed below\n\n#### Explanation:\n\nThe tangent lines to\n\n\\#f\\#\n\nand\n\n\\#g\\#\n\nare given by\n\n\\#t_f->(y-f(a))=f_x(a)(x-a)\\#\n\n\\#t_g->(y-g(b))=g_x(b)(x-b)\\#\n\nwhere\n\n\\#f_x,g_x\\#\n\nindicates derivatives with respect to\n\n\\#x\\#\n\n.\n\n the equations read\n\n\\#t_f->y-6 + 10 a^2 = -20 a (x-a)\\#\n\n\\#t_g->y-8 + (b-2)^2 = -2 (b-2) (x-b)\\#\n\nEquating the\n\n\\#y\\#\n\nvalues\n\n\\#2 (3 + 5 a^2 - 10 a x)=4 + b^2 + (4 - 2 b) x\\#\n\nand imposing the equality for all\n\n\\#x\\#\n\nwe obtain the conditions\n\n\\#2 + 10 a^2 - b^2=0,20 a + 4-2 b=0\\#\n\nSolving for\n\n\\#a,b\\#\n\nwe obtain two solutions:\n\n\\#{a -> 1/45 (-10 - sqrt[55]), b -> 2/9 (-1 - sqrt[55])}\\#\n\nand\n\n\\#{a -> 1/45 (-10 + sqrt[55]), b -> 2/9 (-1 + sqrt[55])}\\#\n", "megamath_md": "# Do #f(x) = 6 – 10x^2# and #g(x) = 8 – (x – 2)^2 # share any tangent lines?\n\n##### 2 Answers\n\nYes, they do. The lines are (approximately)\n\n#### Explanation:\n\n**Function #f#**\n\nAt a point\n\n# = (6-10a^2)+(-20a)(x-a)# which can be simplified to\n\n**Function #g#\n**\n\nAt a point\n\n# = (4+4b-b^2)+(4-2b)(x-b)# which can be simplified to\n\n**The lines coincide** when their slopes are the same and their\n\nThis leads to\n\nThe solutions are\n\nUsing the approximations\n\nwe get corresponding values of\n\nUsing these in\n\nThere are two solutions as detailed below\n\n#### Explanation:\n\nThe tangent lines to\n\nthe equations read\n\nEquating the\n\nand imposing the equality for all\n\nSolving for", "clean_html": "<html><body><header><div><div><div><div><div><a>Calculus</a><div><div><div><h5>Science</h5><ul><li><a>Anatomy & Physiology</a></li><li><a>Astronomy</a></li><li><a>Astrophysics</a></li><li><a>Biology</a></li><li><a>Chemistry</a></li><li><a>Earth Science</a></li><li><a>Environmental Science</a></li><li><a>Organic Chemistry</a></li><li><a>Physics</a></li></ul></div><div><h5>Math</h5><ul><li><a>Algebra</a></li><li><a>Calculus</a></li><li><a>Geometry</a></li><li><a>Prealgebra</a></li><li><a>Precalculus</a></li><li><a>Statistics</a></li><li><a>Trigonometry</a></li></ul></div><div><h5>Humanities</h5><ul><li><a>English Grammar</a></li><li><a>U.S. History</a></li><li><a>World History</a></li></ul></div><div><ul><h5>... and beyond</h5><li><a>Socratic Meta</a></li><li><a>Featured Answers</a></li></ul></div></div></div></div><a>Topics</a><span><form><img></form></span></div></div></div></div></header><div><div><article><h1>Do #f(x) = 6 – 10x^2# and #g(x) = 8 – (x – 2)^2 # share any tangent lines?</h1><div><div><span>Calculus Derivatives Tangent Line to a Curve</span></div></div><div><div><h5><span>2 Answers</span></h5><div><div><div><div><div><a><img></a></div><div><div><a>Jim H</a></div><div><span>May 13, 2016</span></div></div></div><div><div><div><div><div><p>Yes, they do. The lines are (approximately) #y=7.741x+7.498# and #y=1.148x+6.033#</p></div></div></div><div><h4>Explanation:</h4><div><div><p><strong>Function #f#</strong></p><p><mathjax>#f(x) = 6-10x^2# so #f'(x) = -20x#</mathjax></p><p>At a point #(a,f(a))# , the equation of the tangent line is</p><p><mathjax>#y=f(a)+f'(a)(x-a)#</mathjax></p><blockquote><p><mathjax># = (6-10a^2)+(-20a)(x-a)# which can be simplified to</mathjax></p></blockquote><p><mathjax># y= - 20ax+10a^2+6#</mathjax></p><p><strong>Function #g# \n #g(x) = 4+4x-x^2# so #g'(x) = 4-2x#</strong></p><p>At a point #(b,g(b))# , the equation of the tangent line is</p><p><mathjax>#y=g(b)+g'(b)(x-b)#</mathjax></p><blockquote><p><mathjax># = (4+4b-b^2)+(4-2b)(x-b)# which can be simplified to</mathjax></p></blockquote><p><mathjax>#y = (4-2b)x +b^2+4#</mathjax></p><p><strong>The lines coincide when their slopes are the same and their #y# intercepts are the same.</strong></p><p><mathjax>#-20a = 4-2b# which implies #b=10a+2#</mathjax></p><p><mathjax>#10a^2+6=b^2+4 = (10a+2)^2+4#</mathjax></p><p>This leads to</p><p><mathjax>#90a^2+40a+2=0#</mathjax></p><p>The solutions are #a=(-10+-sqrt(55))/45# and #b=10a+2#</p><p>Using the approximations #a ~~ -0.0374178# and #a ~~ -0.387027#</p><p>we get corresponding values of #b ~~ 1.87027# and #b ~~ -1.87027#</p><p>Using these in # y= - 20ax+10a^2+6# or #y = (4-2b)x +b^2+4# gets us the lines</p><p><mathjax>#y=1.148x+6.033# and</mathjax></p><p><mathjax>#y=7.741x+7.498#</mathjax></p></div></div></div></div></div></div><div><a>Answer link</a></div></div><div><div><div><div><a><img></a></div><div><div><a>Cesareo R.</a></div><div><span>May 13, 2016</span></div></div></div><div><div><div><div><div><p>There are two solutions as detailed below</p></div></div></div><div><h4>Explanation:</h4><div><div><p>The tangent lines to #f# and #g# are given by \n #t_f->(y-f(a))=f_x(a)(x-a)# \n #t_g->(y-g(b))=g_x(b)(x-b)# where #f_x,g_x# indicates derivatives with respect to #x# . \n the equations read \n #t_f->y-6 + 10 a^2 = -20 a (x-a)# \n #t_g->y-8 + (b-2)^2 = -2 (b-2) (x-b)# \n Equating the #y# values \n #2 (3 + 5 a^2 - 10 a x)=4 + b^2 + (4 - 2 b) x# \n and imposing the equality for all #x# we obtain the conditions \n #2 + 10 a^2 - b^2=0,20 a + 4-2 b=0# \n Solving for #a,b# we obtain two solutions: \n #{a -> 1/45 (-10 - sqrt[55]), b -> 2/9 (-1 - sqrt[55])}# and \n #{a -> 1/45 (-10 + sqrt[55]), b -> 2/9 (-1 + sqrt[55])}#</p></div></div></div></div></div></div><div><a>Answer link</a></div></div></div></div><div><div><div><h5>Related questions</h5><ul><li><div><a>How do you find the equation of a tangent line to a curve?</a></div></li><li><div><a>How do you find the slope of the tangent line to a curve at a point?</a></div></li><li><div><a>How do you find the tangent line to the curve #y=x^3-9x# at the point where #x=1#?</a></div></li><li><div><a>How do you know if a line is tangent to a curve?</a></div></li><li><div><a>How do you show a line is a tangent to a curve?</a></div></li><li><div><a>How do you find the Tangent line to a curve by implicit differentiation?</a></div></li><li><div><a>What is the slope of a line tangent to the curve #3y^2-2x^2=1#?</a></div></li><li><div><a>How does tangent slope relate to the slope of a line?</a></div></li><li><div><a>What is the slope of a horizontal tangent line?</a></div></li><li><div><a>How do you find the slope of a tangent line using secant lines?</a></div></li></ul><div><a>See all questions in Tangent Line to a Curve</a></div></div><div><div><h5>Impact of this question</h5><div>1611 views around the world</div></div><div><img><div>You can reuse this answer \n Creative Commons License</div></div></div></div></div></div></article></div></div></body></html>"}
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src=\"/resources/lib/html5shiv/html5shiv.js\"></script><![endif]-->\n</head>\n<body class=\"mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-130 ns-subject page-Science_Math_Exam_Resources_Courses_MATH100_December_2010_Question_01_a rootpage-Science_Math_Exam_Resources skin-vector action-view skin-vector-legacy\">\n<div id=\"mw-page-base\" class=\"noprint\"></div>\n<div id=\"mw-head-base\" class=\"noprint\"></div>\n<div id=\"content\" class=\"mw-body\" role=\"main\">\n\t<a id=\"top\"></a>\n\t<div id=\"siteNotice\" class=\"mw-body-content\"></div>\n\t<div class=\"mw-indicators mw-body-content\">\n\t</div>\n\t<h1 id=\"firstHeading\" class=\"firstHeading\" lang=\"en\"><span style=\"/* attempt to bypass $wgRestrictDisplayTitle */\">Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (a)</span></h1>\n\t<div id=\"bodyContent\" class=\"mw-body-content\">\n\t\t<div id=\"siteSub\" class=\"noprint\">From UBC Wiki</div>\n\t\t<div id=\"contentSub\"><span class=\"subpages\">< <a href=\"/Science:Math_Exam_Resources\" title=\"Science:Math Exam Resources\">Science:Math Exam Resources</a>‎ | <a href=\"/Science:Math_Exam_Resources/Courses/MATH100\" title=\"Science:Math Exam Resources/Courses/MATH100\">Courses/MATH100</a>‎ | <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010\">December 2010</a></span></div>\n\t\t<div id=\"contentSub2\"></div>\n\t\t\n\t\t<div id=\"jump-to-nav\"></div>\n\t\t<a class=\"mw-jump-link\" href=\"#mw-head\">Jump to navigation</a>\n\t\t<a class=\"mw-jump-link\" href=\"#searchInput\">Jump to search</a>\n\t\t<div id=\"mw-content-text\" lang=\"en\" dir=\"ltr\" class=\"mw-content-ltr\"><div class=\"mw-parser-output\"><p><br />\n</p>\n<div style=\"font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;\"><b>MATH100</b> <b>December 2010</b></div><p>\n  •  <a class=\"mw-selflink selflink\">Q1 (a)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(b)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (b)\">Q1 (b)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(c)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (c)\">Q1 (c)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(d)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (d)\">Q1 (d)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(e)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (e)\">Q1 (e)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(f)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (f)\">Q1 (f)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(g)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (g)\">Q1 (g)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(h)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (h)\">Q1 (h)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(i)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (i)\">Q1 (i)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(j)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (j)\">Q1 (j)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(k)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (k)\">Q1 (k)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(l)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (l)\">Q1 (l)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(m)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (m)\">Q1 (m)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(n)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (n)\">Q1 (n)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_02_(a)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 02 (a)\">Q2 (a)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_02_(b)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 02 (b)\">Q2 (b)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_02_(c)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 02 (c)\">Q2 (c)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_03\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 03\">Q3</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_04_(a)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (a)\">Q4 (a)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_04_(b)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (b)\">Q4 (b)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_04_(c)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (c)\">Q4 (c)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_04_(d)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (d)\">Q4 (d)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_04_(e)\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (e)\">Q4 (e)</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_05\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 05\">Q5</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_06\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 06\">Q6</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_07\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 07\">Q7</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_08\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010/Question 08\">Q8</a>  •  </p><div class=\"mw-customtoggle-2\" style=\"cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;\">   Other <a href=\"/Science:Math_Exam_Resources/Courses/MATH100\" title=\"Science:Math Exam Resources/Courses/MATH100\"><span style=\"color: white\">MATH100</span></a> Exams</div>\n<div class=\"mw-collapsible\" style=\"\" id=\"mw-customcollapsible-2\"><div class=\"mw-collapsible-content\" style=\"\">  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2011\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2011\">December 2011</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2010\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2010\">December 2010</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2012\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2012\">December 2012</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2013\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2013\">December 2013</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2014\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2014\">December 2014</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2015\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2015\">December 2015</a>  •  <a href=\"/Science:Math_Exam_Resources/Courses/MATH100/December_2016\" title=\"Science:Math Exam Resources/Courses/MATH100/December 2016\">December 2016</a>  •  </div></div>\n<table class=\"collapsible wikitable\" width=\"100%\" style=\"background: #FFFFFF;\">\n<tbody><tr>\n<th style=\"padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;\"><h3 style=\"color:#666666;\"><span id=\"Question_01_(a)\"></span><span class=\"mw-headline\" id=\"Question_01_.28a.29\">Question 01 (a) <font size=\"1\"></font></span></h3>\n</th></tr>\n<tr>\n<td style=\"padding:1em\">\n<p><b>Short-Answer Questions.</b> Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question.\n</p><p><br />\nEvaluate\n</p>\n<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" alttext=\"{\\displaystyle \\displaystyle \\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}}\">\n <semantics>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mstyle displaystyle=\"true\" scriptlevel=\"0\">\n <mstyle displaystyle=\"true\" scriptlevel=\"0\">\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mfrac>\n <mrow>\n <msup>\n <mi>x</mi>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mn>2</mn>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n </mstyle>\n </mstyle>\n </mrow>\n <annotation encoding=\"application/x-tex\">{\\displaystyle \\displaystyle \\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}}</annotation>\n </semantics>\n</math></span><img src=\"https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/355031082e96f21f9bc4c5ada130d318a09de17c\" class=\"mwe-math-fallback-image-inline\" aria-hidden=\"true\" style=\"vertical-align: -2.005ex; width:16.348ex; height:5.843ex;\" alt=\"{\\displaystyle \\displaystyle \\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}}\"/></span></dd></dl>\n<p>Or determine that this limit does not exist.\n</p>\n</td></tr></tbody></table>\n<table width=\"100%\" style=\"background:#ffffff;\">\n<tbody><tr>\n<td style=\"text-align:justify; font-size:8pt; color:#666666;\">\n<p><b>Make sure you understand the problem fully:</b> What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?<br />\n</p>\n</td></tr>\n<tr>\n<td style=\"text-align:justify; font-size:8pt; color:#666666;\">\n<p><b>If you are stuck</b>, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.\n</p>\n</td></tr></tbody></table>\n<p><br />\n</p>\n<table class=\"collapsible wikitable collapsed\" width=\"100%\" style=\"background: #FFFFFF;\">\n<tbody><tr>\n<th style=\"padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;\"><h3 style=\"color:#666666;\"><span class=\"mw-headline\" id=\"Hint_1\">Hint 1 <font size=\"1\"></font></span></h3>\n</th></tr>\n<tr>\n<td style=\"padding:1em\">\n<p>The first step to evaluating a limit is to plug in the value you are limiting to, in this case <i>x</i>=1. This will fail - can you manipulate the numerator and the denominator to handle why it fails?\n</p>\n</td></tr></tbody></table>\n<p><br />\n</p>\n<table class=\"collapsible wikitable collapsed\" width=\"100%\" style=\"background: #FFFFFF;\">\n<tbody><tr>\n<th style=\"padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;\"><h3 style=\"color:#666666;\"><span class=\"mw-headline\" id=\"Hint_2\">Hint 2 <font size=\"1\"></font></span></h3>\n</th></tr>\n<tr>\n<td style=\"padding:1em\">\n<p>Try factoring the numerator.\n</p>\n</td></tr></tbody></table>\n<table width=\"100%\" style=\"background:#ffffff;\">\n<tbody><tr>\n<td style=\"text-align:justify; font-size:8pt; color:#666666;\">\n<p>Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.\n</p>\n<ul><li><b>If you are stuck on a problem:</b> Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.</li>\n<li><b>If you want to check your work:</b> Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.<br /></li></ul>\n</td></tr></tbody></table>\n<p><br />\n</p>\n<table class=\"collapsible wikitable collapsed\" width=\"100%\" style=\"background: #FFFFFF;\">\n<tbody><tr>\n<th style=\"padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;\"><h3 style=\"color:#666666;\"><span class=\"mw-headline\" id=\"Solution_1\">Solution 1 <font size=\"1\"></font></span></h3>\n</th></tr>\n<tr>\n<td style=\"padding:1em\"><div style=\"color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;\"><span class=\"plainlinks\">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? <a rel=\"nofollow\" class=\"external text\" href=\"https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578\">Let us know here</a>.<br /> Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>\n<p>Directly plugging in <i>x</i>=1 into the limit gives 0/0. We try to factor to cancel out the terms making the fraction zero.\n</p>\n<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" alttext=\"{\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x-1)(x+3)}{x-1}}\\\\=&\\lim _{x\\to 1}(x+3)\\\\=&(1)+3\\\\=&4\\end{aligned}}}\">\n <semantics>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mstyle displaystyle=\"true\" scriptlevel=\"0\">\n <mstyle displaystyle=\"true\" scriptlevel=\"0\">\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\">\n <mtr>\n <mtd />\n <mtd>\n <mi></mi>\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mfrac>\n <mrow>\n <msup>\n <mi>x</mi>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mn>2</mn>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mi></mi>\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mfrac>\n <mrow>\n <mo stretchy=\"false\">(</mo>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>1</mn>\n <mo stretchy=\"false\">)</mo>\n <mo stretchy=\"false\">(</mo>\n <mi>x</mi>\n <mo>+</mo>\n <mn>3</mn>\n <mo stretchy=\"false\">)</mo>\n </mrow>\n <mrow>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mi></mi>\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mo stretchy=\"false\">(</mo>\n <mi>x</mi>\n <mo>+</mo>\n <mn>3</mn>\n <mo stretchy=\"false\">)</mo>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mi></mi>\n <mo stretchy=\"false\">(</mo>\n <mn>1</mn>\n <mo stretchy=\"false\">)</mo>\n <mo>+</mo>\n <mn>3</mn>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mn>4</mn>\n </mtd>\n </mtr>\n </mtable>\n </mrow>\n </mstyle>\n </mstyle>\n </mrow>\n <annotation encoding=\"application/x-tex\">{\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x-1)(x+3)}{x-1}}\\\\=&\\lim _{x\\to 1}(x+3)\\\\=&(1)+3\\\\=&4\\end{aligned}}}</annotation>\n </semantics>\n</math></span><img src=\"https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d9167c93b42422e3e39b40697bfdaaca991840b7\" class=\"mwe-math-fallback-image-inline\" aria-hidden=\"true\" style=\"vertical-align: -10.505ex; width:21.859ex; height:22.176ex;\" alt=\"{\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x-1)(x+3)}{x-1}}\\\\=&\\lim _{x\\to 1}(x+3)\\\\=&(1)+3\\\\=&4\\end{aligned}}}\"/></span></dd></dl>\n</td></tr></tbody></table>\n<p><br />\n</p>\n<table class=\"collapsible wikitable collapsed\" width=\"100%\" style=\"background: #FFFFFF;\">\n<tbody><tr>\n<th style=\"padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;\"><h3 style=\"color:#666666;\"><span class=\"mw-headline\" id=\"Solution_2\">Solution 2 <font size=\"1\"></font></span></h3>\n</th></tr>\n<tr>\n<td style=\"padding:1em\"><div style=\"color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;\"><span class=\"plainlinks\">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? <a rel=\"nofollow\" class=\"external text\" href=\"https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578\">Let us know here</a>.<br /> Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>\n<p>Directly plugging in <i>x</i>=1 into the limit gives 0/0. Since this is an indeterminate form, we can use L'Hospital's rule:\n</p>\n<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" alttext=\"{\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x^{2}+2x-3)'}{(x-1)'}}\\\\=&\\lim _{x\\to 1}{\\frac {2x+2}{1}}\\\\=&\\lim _{x\\to 1}{2x+2}\\\\=&2(1)+2\\\\=&4\\end{aligned}}}\">\n <semantics>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mstyle displaystyle=\"true\" scriptlevel=\"0\">\n <mstyle displaystyle=\"true\" scriptlevel=\"0\">\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\">\n <mtr>\n <mtd />\n <mtd>\n <mi></mi>\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mfrac>\n <mrow>\n <msup>\n <mi>x</mi>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mn>2</mn>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mi></mi>\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mfrac>\n <mrow>\n <mo stretchy=\"false\">(</mo>\n <msup>\n <mi>x</mi>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mn>2</mn>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>3</mn>\n <msup>\n <mo stretchy=\"false\">)</mo>\n <mo>′</mo>\n </msup>\n </mrow>\n <mrow>\n <mo stretchy=\"false\">(</mo>\n <mi>x</mi>\n <mo>−<!-- − --></mo>\n <mn>1</mn>\n <msup>\n <mo stretchy=\"false\">)</mo>\n <mo>′</mo>\n </msup>\n </mrow>\n </mfrac>\n </mrow>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mi></mi>\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mfrac>\n <mrow>\n <mn>2</mn>\n <mi>x</mi>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n <mn>1</mn>\n </mfrac>\n </mrow>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mi></mi>\n <munder>\n <mo movablelimits=\"true\" form=\"prefix\">lim</mo>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mi>x</mi>\n <mo stretchy=\"false\">→<!-- → --></mo>\n <mn>1</mn>\n </mrow>\n </munder>\n <mrow class=\"MJX-TeXAtom-ORD\">\n <mn>2</mn>\n <mi>x</mi>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mn>2</mn>\n <mo stretchy=\"false\">(</mo>\n <mn>1</mn>\n <mo stretchy=\"false\">)</mo>\n <mo>+</mo>\n <mn>2</mn>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <mo>=</mo>\n </mtd>\n <mtd>\n <mn>4</mn>\n </mtd>\n </mtr>\n </mtable>\n </mrow>\n </mstyle>\n </mstyle>\n </mrow>\n <annotation encoding=\"application/x-tex\">{\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x^{2}+2x-3)'}{(x-1)'}}\\\\=&\\lim _{x\\to 1}{\\frac {2x+2}{1}}\\\\=&\\lim _{x\\to 1}{2x+2}\\\\=&2(1)+2\\\\=&4\\end{aligned}}}</annotation>\n </semantics>\n</math></span><img src=\"https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/cbabc2cb50106de485951ea02a534c5f23e9d1a7\" class=\"mwe-math-fallback-image-inline\" aria-hidden=\"true\" style=\"vertical-align: -13.671ex; width:21.789ex; height:28.509ex;\" alt=\"{\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x^{2}+2x-3)'}{(x-1)'}}\\\\=&\\lim _{x\\to 1}{\\frac {2x+2}{1}}\\\\=&\\lim _{x\\to 1}{2x+2}\\\\=&2(1)+2\\\\=&4\\end{aligned}}}\"/></span></dd></dl>\n</td></tr></tbody></table>\n<p><br />\n</p>\n<div class=\"mw-customtoggle-11\" style=\"cursor:pointer;background-color:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:center\"> Click here for similar questions </div>\n<div class=\"mw-collapsible mw-collapsed\" style=\"\" id=\"mw-customcollapsible-11\"><div class=\"mw-collapsible-content\" style=\"\"><div id=\"MER-tag\">MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Limit of a rational function, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag</div></div></div>\n<div class=\"noprint\" style=\"float:none; border:5px solid #7491A3;width:280px; padding:10px;\">\n<table cellspacing=\"0\">\n<tbody><tr>\n<td><div class=\"floatnone\"><a href=\"/File:Bulbgraph.png\" 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Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question. Evaluate ${\\displaystyle \\displaystyle \\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}}$ Or determine that this limit does not exist. |\n\n| Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? |\n|---|\n| If you are stuck , check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. |\n\n| Hint 1 |\n|---|\n| The first step to evaluating a limit is to plug in the value you are limiting to, in this case x =1. This will fail - can you manipulate the numerator and the denominator to handle why it fails? |\n\n| Hint 2 |\n|---|\n| Try factoring the numerator. |\n\nChecking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.\n\n| Solution 1 |\n|---|\n| Found a typo ? Is this solution unclear ? Let us know here . Please rate my easiness ! It's quick and helps everyone guide their studies. Directly plugging in x =1 into the limit gives 0/0. We try to factor to cancel out the terms making the fraction zero. ${\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x-1)(x+3)}{x-1}}\\\\=&\\lim _{x\\to 1}(x+3)\\\\=&(1)+3\\\\=&4\\end{aligned}}}$ |\n\n| Solution 2 |\n|---|\n| Found a typo ? Is this solution unclear ? Let us know here . Please rate my easiness ! It's quick and helps everyone guide their studies. Directly plugging in x =1 into the limit gives 0/0. Since this is an indeterminate form, we can use L'Hospital's rule: ${\\displaystyle \\displaystyle {\\begin{aligned}&\\lim _{x\\to 1}{\\frac {x^{2}+2x-3}{x-1}}\\\\=&\\lim _{x\\to 1}{\\frac {(x^{2}+2x-3)'}{(x-1)'}}\\\\=&\\lim _{x\\to 1}{\\frac {2x+2}{1}}\\\\=&\\lim _{x\\to 1}{2x+2}\\\\=&2(1)+2\\\\=&4\\end{aligned}}}$ |\n", "megamath_md": "Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (a)\n• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q4 (e) • Q5 • Q6 • Q7 • Q8 •\nQuestion 01 (a) \n\nShortAnswer Questions. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question.\nOr determine that this limit does not exist. \nMake sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? \nIf you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. \nHint 1 \n\nThe first step to evaluating a limit is to plug in the value you are limiting to, in this case x=1. This will fail can you manipulate the numerator and the denominator to handle why it fails? \nHint 2 \n\nTry factoring the numerator. \nChecking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.\n\nSolution 1 \n\nFound a typo? Is this solution unclear? Let us know here.\nPlease rate my easiness! It's quick and helps everyone guide their studies. Directly plugging in x=1 into the limit gives 0/0. We try to factor to cancel out the terms making the fraction zero. \nSolution 2 \n\nFound a typo? Is this solution unclear? Let us know here.\nPlease rate my easiness! It's quick and helps everyone guide their studies. Directly plugging in x=1 into the limit gives 0/0. Since this is an indeterminate form, we can use L'Hospital's rule: ", "clean_html": "<html><body><div><h1><span>Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (a)</span></h1><div><div>From UBC Wiki</div><div><span>< Science:Math Exam Resources | Courses/MATH100 | December 2010</span></div><a>Jump to navigation Jump to search</a><div><div><p><br></p><div><b>MATH100 December 2010</b></div><p> • Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q4 (e) • Q5 • Q6 • Q7 • Q8 • </p><div> Other MATH100 Exams</div><div><div> • December 2011 • December 2010 • December 2012 • December 2013 • December 2014 • December 2015 • December 2016 • </div></div><table><tbody><tr><th><h3><span>Question 01 (a)</span></h3></th></tr><tr><td><p><b>Short-Answer Questions. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question.</b></p><p><span>\n Evaluate</span></p><dl><dd><span><img></span></dd></dl><p>Or determine that this limit does not exist.</p></td></tr></tbody></table><table><tbody><tr><td><p><b>Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? \n</b></p></td></tr><tr><td><p><b>If you are stuck , check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.</b></p></td></tr></tbody></table><p><br></p><table><tbody><tr><th><h3><span>Hint 1</span></h3></th></tr><tr><td><p>The first step to evaluating a limit is to plug in the value you are limiting to, in this case x =1. This will fail - can you manipulate the numerator and the denominator to handle why it fails?</p></td></tr></tbody></table><p><br></p><table><tbody><tr><th><h3><span>Hint 2</span></h3></th></tr><tr><td><p>Try factoring the numerator.</p></td></tr></tbody></table><table><tbody><tr><td><p>Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.</p><ul><li><b>If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.</b></li><li><b>If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result. \n</b></li></ul></td></tr></tbody></table><p><br></p><table><tbody><tr><th><h3><span>Solution 1</span></h3></th></tr><tr><td><div><span>Found a typo? Is this solution unclear? Let us know here.\nPlease rate my easiness! It's quick and helps everyone guide their studies.</span></div><p>Directly plugging in x =1 into the limit gives 0/0. We try to factor to cancel out the terms making the fraction zero.</p><dl><dd><span><img></span></dd></dl></td></tr></tbody></table><p><br></p><table><tbody><tr><th><h3><span>Solution 2</span></h3></th></tr><tr><td><div><span>Found a typo? Is this solution unclear? Let us know here.\nPlease rate my easiness! It's quick and helps everyone guide their studies.</span></div><p>Directly plugging in x =1 into the limit gives 0/0. Since this is an indeterminate form, we can use L'Hospital's rule:</p><dl><dd><span><img></span></dd></dl></td></tr></tbody></table><p><br></p><div>Click here for similar questions</div><div><div><div>MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Limit of a rational function, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag</div></div></div><div><table><tbody><tr><td><div><a><img></a></div></td><td><b>Math Learning Centre</b><ul><li>A space to study math together.</li><li>Free math graduate and undergraduate TA support.</li><li>Mon - Fri: 11 am - 5 pm, LSK 301&302.</li></ul><p><b>Private tutor</b></p><ul><li>We can help to find a private tutor.</li></ul></td></tr></tbody></table></div><p><br></p></div></div><div>Retrieved from \" https://wiki.ubc.ca/index.php?title=Science:Math_Exam_Resources/Courses/MATH100/December_2010/Question_01_(a)&oldid=357222 \"</div><div><div><a>Categories :</a><ul><li><a>Pages using DynamicPageList parser function</a></li><li><a>Pages using DynamicPageList parser tag</a></li></ul></div><div>Hidden categories:<ul><li><a>MER QGQ flag</a></li><li><a>MER QGH flag</a></li><li><a>MER QGS flag</a></li><li><a>MER RT flag</a></li><li><a>MER Tag Limit of a rational function</a></li></ul></div></div></div></div><div><h2>Navigation menu</h2></div></body></html>"}