// Numbas version: exam_results_page_options
{"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": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["w", "stat1", "b3", "a3", "a2", "b2", "v", "c2", "d2", "stat2"], "functions": {"plotf": {"language": "javascript", "parameters": [["a2", "number"], ["b2", "number"], ["c2", "number"], ["a3", "number"], ["b3", "number"], ["stat1", "number"], ["stat2", "number"]], "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;", "type": "html"}}, "extensions": ["jsxgraph"], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "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>", "variables": {"c2": {"templateType": "anything", "description": "", "definition": "random(1..5)", "group": "Ungrouped variables", "name": "c2"}, "a3": {"templateType": "anything", "description": "", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "name": "a3"}, "v": {"templateType": "anything", "description": "", "definition": "a3*d2+b3", "group": "Ungrouped variables", "name": "v"}, "w": {"templateType": "anything", "description": "", "definition": "a2*d2^2+2*a2*b2*d2+c2+a2*b2^2", "group": "Ungrouped variables", "name": "w"}, "b2": {"templateType": "anything", "description": "", "definition": "-random(1..6)", "group": "Ungrouped variables", "name": "b2"}, "d2": {"templateType": "anything", "description": "", "definition": "random(0..5)", "group": "Ungrouped variables", "name": "d2"}, "stat1": {"templateType": "anything", "description": "", "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)", "group": "Ungrouped variables", "name": "stat1"}, "b3": {"templateType": "anything", "description": "", "definition": "random(-9..9)", "group": "Ungrouped variables", "name": "b3"}, "a2": {"templateType": "anything", "description": "", "definition": "random(1..3)", "group": "Ungrouped variables", "name": "a2"}, "stat2": {"templateType": "anything", "description": "", "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)", "group": "Ungrouped variables", "name": "stat2"}}, "tags": [], "parts": [{"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "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>", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "notallowed": {"message": "<p>Enter as a fraction or an integer and not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "checkingAccuracy": 0.001, "scripts": {}, "answer": "{v}/{w}", "valuegenerators": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "all, fractionNumbers", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}], "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "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>", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "({a3}*a+{b3})/({a2}*a^2+{2*a2*b2}*a+{c2+a2*b2^2})", "valuegenerators": [{"value": "", "name": "a"}], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "all", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "showCellAnswerState": true, "maxMarks": 0, "scripts": {}, "minMarks": 0, "matrix": [0, 1, 0, 0, 0], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "shuffleChoices": true, "showFeedbackIcon": true, "marks": 0, "displayType": "radiogroup", "unitTests": [], "distractors": ["Not true as $f(t)=0$ for a value of $t$.", "", "", "", ""], "displayColumns": 0, "useCustomName": false, "variableReplacements": [], "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>"]}], "variableReplacements": []}], "rulesets": {}, "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>", "name": "Find the limit of an algebraic fraction as parameter tends to a given value", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}