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{"name": "Use algebra to evaluate limit 5", "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 5", "tags": [], "metadata": {"description": "<p>Evaluate a rational limit using algebra</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate a rational limit using algebra</p>", "advice": "<p>Use the difference of squares formula</p>\n<p>\\[ a^2 - b^2 = (a-b)(a+b) \\]</p>\n<p>to simplify the denominator.</p>\n<p>So, \\[ x-\\var{xlim} = (\\sqrt{x}-\\sqrt{\\var{xlim}})(\\sqrt{x}+\\sqrt{\\var{xlim}}) \\] and the limit becomes</p>\n<p>\\[ \\lim_{x\\to \\var{xlim}} \\frac{\\sqrt{x}-\\sqrt{\\var{xlim}}}{(\\sqrt{x}-\\sqrt{\\var{xlim}})(\\sqrt{x}+\\sqrt{\\var{xlim}})} =&nbsp; \\lim_{x\\to \\var{xlim}} \\frac{1}{(x+\\sqrt{\\var{xlim}})} = \\frac{1}{2*\\sqrt{\\var{xlim}}} \\]</p>", "rulesets": {}, "extensions": [], "variables": {"xlim": {"name": "xlim", "group": "Ungrouped variables", "definition": "random(1..7)", "description": "<p>x limit</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["xlim"], "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{xlim}} \\frac{\\sqrt{x}-\\simplify{sqrt({xlim})} }{x-\\var{xlim}}.\\]</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": "1/(2*sqrt({xlim}))", "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/"}]}