// Numbas version: exam_results_page_options {"name": "Evaluate a limit #1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Evaluate a limit #1", "tags": [], "metadata": {"description": "

Evaluate a limit (x-a)/(x^2+(a+b)x+ab)

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Evaluate the following limit

\n

\n
", "advice": "

\\(\\displaystyle{\\lim_{x \\to \\var{a}}}\\frac{x-\\var{a}}{x^2-\\simplify{{a}+{b}}x+\\simplify{{a}*{b}}}\\)

\n

We can factorise the denominator

\n

\\(\\displaystyle{\\lim_{x \\to \\var{a}}}\\frac{x-\\var{a}}{(x-\\var{a})(x-\\var{b})}\\)

\n

Cancel out the common factor

\n

\\(\\displaystyle{\\lim_{x \\to \\var{a}}}\\frac{1}{x-\\var{b}}\\)

\n

Insert the value \\(\\var{a}\\) in for \\(x\\) to evaluate the limit

\n

\\(-\\frac{1}{\\simplify{{b}-{a}}}\\)

\n

", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(6 .. 15#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "a$ \\lim_{x \\to \\var{a}}$  $\\frac{x-\\var{a}}{ \\simplify{x^2-({a}+{b})x+{a}*{b} } } =$[[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/({a}-{b})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Yuri Anissimov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1705/"}, {"name": "Nathan Garland", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14584/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Yuri Anissimov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1705/"}, {"name": "Nathan Garland", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14584/"}]}