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{"name": "Use algebra to evaluate limit 7", "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 7", "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>\\[ \\lim_{x\\to \\var{xlim}} \\frac{\\simplify{{xlim}*x^{nnum}-x^{{nnum}+1}}}{\\sqrt{\\var{xlim}}-\\sqrt{x}}.\\]</p>\n<p>Start by factorising the numerator.&nbsp;</p>\n<p>\\[ \\lim_{x\\to \\var{xlim}} \\frac{\\simplify{x^{nnum}({xlim}-x)}}{\\sqrt{\\var{xlim}}-\\sqrt{x}}.\\]</p>\n<p>Then apply difference of squares&nbsp;&nbsp;$a^2 - b^2 = (a-b)(a+b)$ to the second term in the numerator with $a=\\sqrt{\\var{xlim}}$ and $b=\\sqrt{x}$. This gives</p>\n<p>\\[ \\lim_{x\\to \\var{xlim}} \\frac{\\simplify{x^{nnum}({xlim}-x)}}{\\sqrt{\\var{xlim}}-\\sqrt{x}} =\\lim_{x\\to \\var{xlim}}&nbsp;x^{\\var{nnum}}(\\sqrt{\\var{xlim}}+\\sqrt{x}). \\]</p>\n<p>We can now substitute the limit to obtain $\\simplify{2*sqrt({xlim})*{xlim}^{nnum}}$.</p>\n<p>PS: Remember to use the <code>sqrt()</code> command to enter radicals if you need to.</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"xlim": {"name": "xlim", "group": "Ungrouped variables", "definition": "random(1..7)", "description": "<p>x limit</p>", "templateType": "anything"}, "nnum": {"name": "nnum", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["xlim", "nnum"], "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{\\simplify{{xlim}*x^{nnum}-x^{{nnum}+1}}}{\\sqrt{\\var{xlim}}-\\sqrt{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": "2*sqrt({xlim})*{xlim}^{nnum}", "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/"}]}