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{"name": "Find the limit of an algebraic fraction as parameter tends to a given value", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a2*d2^2+2*a2*b2*d2+c2+a2*b2^2", "name": "w", "description": ""}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..5)", "name": "d2", "description": ""}, "b3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)", "name": "b3", "description": ""}, "stat2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(2*a2*b3-sqrt(4*a2^2*b3^2+4*a3*a2*(a3*c2-2*b3*b2*a2+a2*b2^2*a3)))/(-2*a3*a2)", "name": "stat2", "description": ""}, "a3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "name": "a3", "description": ""}, "stat1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(2*a2*b3+sqrt(4*a2^2*b3^2+4*a3*a2*(a3*c2-2*b3*b2*a2+a2*b2^2*a3)))/(-2*a3*a2)", "name": "stat1", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-random(1..6)", "name": "b2", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a3*d2+b3", "name": "v", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "a2", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "c2", "description": ""}}, "ungrouped_variables": ["w", "stat1", "b3", "a3", "a2", "b2", "v", "c2", "d2", "stat2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find the limit of an algebraic fraction as parameter tends to a given value", "functions": {"plotf": {"type": "html", "language": "javascript", "definition": "var f = function(t){ return (a3*t+b3)/(a2*t*t+2*a2*b2*t+c2+a2*b2*b2); };\nvar m1=Math.min(stat1,stat2);\nvar m2=Math.max(stat1,stat2);\nvar f1=f(stat1);\nvar f2=f(stat2);\nvar a=Math.abs(f1);\nvar b=Math.abs(f2);\nvar M=Math.max(a,b);\nvar div = Numbas.extensions.jsxgraph.makeBoard('300px','300px', {axis:true,showNavigation:false,boundingbox:[m1-10,M+2,m2+10,-M-2]});\n\nvar brd=div.board;\n\nvar plot = brd.create('functiongraph',[f,m1-10,m2+10]);\n\n//brd.create('text',[c,-2,c]);\n//var i1 = brd.create('integral', [[0, c], plot]);\n\nreturn div;", "parameters": [["a2", "number"], ["b2", "number"], ["c2", "number"], ["a3", "number"], ["b3", "number"], ["stat1", "number"], ["stat2", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{v}/{w}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Enter as a fraction or an integer and not as a decimal.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "all, fractionNumbers", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Find the following limit:</p>\n<p>$\\displaystyle \\simplify{Limit(f(t),t,{d2}) }= \\;$[[0]].</p>\n<p>Enter your answer as a fraction or an integer and not as a decimal.</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "({a3}*a+{b3})/({a2}*a^2+{2*a2*b2}*a+{c2+a2*b2^2})", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Because $f(t) \\neq 0, \\; \\forall t \\in \\mathbb{R}$.</p>", "<p>Because $f$ is continuous at all points in $\\mathbb{R}$.</p>", "<p>Because $f$ is a function defined in terms of polynomials.</p>", "<p>Because all ratios of polynomials are continuous.</p>", "<p>Because $f$ is differentiable at all points.</p>"], "matrix": [0, 1, 0, 0, 0], "distractors": ["Not true as $f(t)=0$ for a value of $t$.", "", "", "", ""], "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "marks": 0}], "type": "gapfill", "prompt": "<p>Also find:</p>\n<p>$\\displaystyle \\simplify{Limit(f(t),t,a) }= \\;$[[0]], where $a \\in \\mathbb{R}$ is any point.</p>\n<p>Why can we evaluate this limit? [[1]]</p>", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Let the function $f$ be given by $\\displaystyle f(t)=\\simplify{({a3} * t + {b3}) / ({a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2}) }$</p>", "tags": ["checked2015", "MAS2224"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": "<p><strong>Graph of $f$.</strong></p>\n<p>{plotf(a2,b2,c2,a3,b3,stat1,stat2)}</p>\n<p></p>\n<p>Note that $\\simplify{{a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2} ={a2}*(t+{b2})^2+{c2}} \\gt 0$.</p>\n<p>Hence the denominator of $f(t) \\neq 0,\\;\\forall t \\in \\mathbb{R}$ and so $f$ is continuous at all points in $\\mathbb{R}$.</p>\n<p>This means that in part a) we can take the limit by simply subsituting $t=\\var{d2}$ into the expression for $f(t)$ and we get:</p>\n<p>\\[\\lim_{x \\to \\var{d2}}f(t)=\\simplify[all,!otherNumbers,fractionNumbers,!collectNumbers]{({a3} * {d2} + {b3}) / ({a2} * {d2}^ 2 + {2 * b2 * a2} * {d2}+ {c2 + a2 * b2 ^ 2})={v}/{w} }\\]</p>\n<p>Similarly in part b) we have :</p>\n<p>\\[\\lim_{x \\to a}f(t)=\\simplify[all,!collectNumbers,!otherNumbers,fractionNumbers]{({a3} * a + {b3}) / ({a2} * a^ 2 + {2 * b2 * a2} * a+ {c2 + a2 * b2 ^ 2})}\\]</p>\n<p></p>\n<p>As noted above we can find this limit by simply putting $t=a$ into the formula for the function as $f$ is continuous at all points in $\\mathbb{R}$.</p>\n<p></p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}