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{"name": "Use algebra to evaluate limit 6", "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 6", "tags": [], "metadata": {"description": "<p>Evaluate&nbsp;$\\displaystyle \\lim_{x\\to 0} \\frac{\\sqrt{ax+b} - d}{cx}$&nbsp;using algebra</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate&nbsp;$\\displaystyle \\lim_{x\\to 0} \\frac{\\sqrt{ax+b} - d}{cx}$&nbsp;using algebra</p>", "advice": "<p>Rationalise expression so that $x$ cancels.</p>\n<p>Note that $\\displaystyle \\frac{\\sqrt{\\simplify{{anum}*x+{bnum}}}-\\var{dnum}}{\\var{cnum}x} =&nbsp;\\frac{\\sqrt{\\simplify{{anum}*x+{bnum}}}-\\var{dnum}}{\\var{cnum}x}\\cdot&nbsp;\\frac{\\sqrt{\\simplify{{anum}*x+{bnum}}}+\\var{dnum}}{\\sqrt{\\simplify{{anum}*x+{bnum}}}+\\var{dnum}} = \\frac{\\simplify{{anum}*x+{bnum}} -\\simplify{{dnum}*{dnum}}}{\\var{cnum}x(\\sqrt{\\simplify{{anum}*x+{bnum}}}+\\var{dnum})}$.</p>\n<p>This simplifies to $\\displaystyle \\frac{\\var{anum}}{\\var{cnum}(\\sqrt{\\simplify{{anum}*x+{bnum}}}+\\var{dnum})}$.</p>\n<p>Hence,</p>\n<p>\\[ \\lim_{x\\to 0} \\frac{\\sqrt{\\simplify{{anum}*x+{bnum}}}-\\var{dnum}}{\\var{cnum}x} = \\lim_{x\\to 0}&nbsp;\\frac{\\var{anum}}{\\var{cnum}(\\sqrt{\\simplify{{anum}*x+{bnum}}}+\\var{dnum})} =&nbsp; \\frac{\\var{anum}}{\\simplify{2*{cnum}*{dnum}}}. \\]</p>", "rulesets": {}, "extensions": [], "variables": {"anum": {"name": "anum", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "<p>constant a</p>", "templateType": "anything"}, "bnum": {"name": "bnum", "group": "Ungrouped variables", "definition": "random(1,4,9,16,25)", "description": "<p>perfect square</p>", "templateType": "anything"}, "dnum": {"name": "dnum", "group": "Ungrouped variables", "definition": "sqrt(bnum)", "description": "", "templateType": "anything"}, "cnum": {"name": "cnum", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>constant c</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["anum", "bnum", "dnum", "cnum"], "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 0} \\frac{\\sqrt{\\simplify{{anum}*x+{bnum}}}-\\var{dnum} }{\\simplify{{cnum}*x}}.\\]</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": "{anum}/(2*{cnum}*{dnum})", "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/"}]}