// Numbas version: exam_results_page_options
{"name": "Use algebra to evaluate limit 8", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Use algebra to evaluate limit 8", "tags": [], "metadata": {"description": "<p>Evaluate&nbsp;$\\displaystyle \\lim_{x\\to k} \\frac{x+a}{\\sqrt{x+b} - c}$&nbsp;using algebra</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate&nbsp;$\\displaystyle \\lim_{x\\to k} \\frac{x+a}{\\sqrt{x+b} - c}$&nbsp;using algebra</p>", "advice": "<p>Rationalise expression taking advantage of difference of squares formula $a^2-b^2=(a-b)(a+b)$.</p>\n<p>Note that $\\displaystyle&nbsp;\\frac{\\simplify{x-{anum}}}{\\sqrt{\\simplify{x+{dnum}}}-\\var{bnum}}=\\frac{\\simplify{x-{anum}}}{\\sqrt{\\simplify{x+{dnum}}}-\\var{bnum}}\\cdot\\frac{\\sqrt{\\simplify{x+{dnum}}}+\\var{bnum}}{\\sqrt{\\simplify{x+{dnum}}}+\\var{bnum}} = \\frac{(\\simplify{x-{anum}})({\\sqrt{\\simplify{x+{dnum}}}+\\var{bnum}})}{x+\\var{dnum}-\\var{bs}} $, which simplifies to ${\\sqrt{\\simplify{x+{dnum}}}+\\var{bnum}}$.</p>\n<p>Hence,</p>\n<p>\\[ \\lim_{x\\to \\var{anum}} \\frac{\\simplify{x-{anum}}}{\\sqrt{\\simplify{x+{dnum}}}-\\var{bnum} } =&nbsp;\\lim_{x\\to \\var{anum}}&nbsp;{\\sqrt{\\simplify{x+{dnum}}}+\\var{bnum}} = \\simplify{2*{bnum}}.\\]</p>", "rulesets": {}, "extensions": [], "variables": {"anum": {"name": "anum", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>constant a</p>", "templateType": "anything"}, "bnum": {"name": "bnum", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "<p>constant b</p>", "templateType": "anything"}, "dnum": {"name": "dnum", "group": "Ungrouped variables", "definition": "bs-anum", "description": "", "templateType": "anything"}, "cnum": {"name": "cnum", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>constant c</p>", "templateType": "anything"}, "bs": {"name": "bs", "group": "Ungrouped variables", "definition": "bnum*bnum", "description": "<p>b squared</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["anum", "bnum", "dnum", "cnum", "bs"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Evaluate</p>\n<p>\\[ \\lim_{x\\to \\var{anum}} \\frac{\\simplify{x-{anum}}}{\\sqrt{\\simplify{x+{dnum}}}-\\var{bnum} }.\\]</p>\n<p>giving your answer in exact form.&nbsp;</p>\n<p><strong>Answer</strong>:&nbsp;[[0]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "2*{bnum}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}]}], "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}