// Numbas version: exam_results_page_options
{"name": "Limits using algebra", "metadata": {"description": "<p>Calculating limits using algebraic techniques</p>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "duration": 0, "percentPass": "50", "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "questions": [{"name": "Use algebra to evaluate limit 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "tags": [], "metadata": {"description": "<p>Evaluate $\\displaystyle \\quad \\lim_{x\\to a} \\frac{ax+b}{x^2+c} \\quad $ algebraically.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate the $\\displaystyle \\quad \\lim_{x\\to a} \\frac{ax+b}{x^2+c} \\quad $ algebraically.</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^2-\\var{xlimsquared} =\\simplify{(x-{xlim})(x+{xlim})} \\] and the limit becomes</p>\n<p>\\[ \\lim_{x\\to \\var{xlim}} \\frac{\\simplify{{x}-{xlim}}}{\\simplify{(x-{xlim})(x+{xlim})}} =&nbsp; \\lim_{x\\to \\var{xlim}} \\frac{1}{\\simplify{(x+{xlim})}} = \\frac{1}{\\simplify{2*{xlim}}} \\]</p>", "rulesets": {}, "variables": {"xlim": {"name": "xlim", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>x limit</p>", "templateType": "anything"}, "xlimsquared": {"name": "xlimsquared", "group": "Ungrouped variables", "definition": "xlim*xlim", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["xlim", "xlimsquared"], "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{x-{xlim}} }{x^2-\\var{xlimsquared}}.\\]</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*{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"}, {"name": "Use algebra to evaluate limit 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "tags": [], "metadata": {"description": "<p>Evaluate $\\displaystyle \\quad \\lim_{x\\to a} \\frac{ax^2+bx+c}{ex+d} \\quad $ algebraically.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate $\\displaystyle \\quad \\lim_{x\\to a} \\frac{ax^2+bx+c}{ex+d} \\quad $ algebraically.</p>", "advice": "<p></p>\n<p>Factorise the numerator to obtain</p>\n<p>\\[ \\lim_{x\\to \\var{anum}} \\frac{\\simplify{{knum}({x}-{anum})(x+{bnum})}}{\\simplify{(x-{anum})}} =\\lim_{x\\to \\var{anum}} \\var{knum}(\\simplify{x+{bnum}}) =\\simplify{{knum}*( {anum}+{bnum})}\\]</p>", "rulesets": {}, "variables": {"anum": {"name": "anum", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>a constant</p>", "templateType": "anything"}, "bnum": {"name": "bnum", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>constant b</p>", "templateType": "anything"}, "bma": {"name": "bma", "group": "Ungrouped variables", "definition": "bnum-anum", "description": "<p>bnum - anum</p>", "templateType": "anything"}, "atb": {"name": "atb", "group": "Ungrouped variables", "definition": "anum*bnum", "description": "<p>anum x bnum</p>", "templateType": "anything"}, "knum": {"name": "knum", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "<p>constant k</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["anum", "bnum", "bma", "atb", "knum"], "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{{knum}*x^2+{knum}*{bma}x-{knum}*{atb}} }{\\simplify{x-{anum}}}.\\]</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": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{knum}*({anum}+{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"}, {"name": "Use algebra to evaluate limit 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "tags": [], "metadata": {"description": "<p>Evaluate $\\displaystyle \\quad \\lim_{x\\to a} \\frac{ex+d}{ax^2+bx+c} \\quad $ algebraically.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate $\\displaystyle \\quad \\lim_{x\\to a} \\frac{ex+d}{ax^2+bx+c} \\quad $ algebraically.</p>", "advice": "<p></p>\n<p>Factorise the denomenator to obtain</p>\n<p>\\[ \\lim_{x\\to \\var{bnum}} \\frac{\\simplify{(x-{bnum})}}{\\simplify{{knum}({x}-{bnum})(x+{anum})}} =\\lim_{x\\to \\var{bnum}} \\frac{1}{\\var{knum}(\\simplify{x+{anum}})} ={\\frac{1}{\\simplify{{knum}*({anum}+{bnum})}}}\\]</p>", "rulesets": {}, "variables": {"anum": {"name": "anum", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>a constant</p>", "templateType": "anything"}, "bnum": {"name": "bnum", "group": "Ungrouped variables", "definition": "random(1..10 except -anum)", "description": "<p>constant b</p>", "templateType": "anything"}, "bma": {"name": "bma", "group": "Ungrouped variables", "definition": "bnum-anum", "description": "<p>bnum - anum</p>", "templateType": "anything"}, "atb": {"name": "atb", "group": "Ungrouped variables", "definition": "anum*bnum", "description": "<p>anum x bnum</p>", "templateType": "anything"}, "knum": {"name": "knum", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "<p>constant k</p>", "templateType": "anything"}, "apb": {"name": "apb", "group": "Ungrouped variables", "definition": "anum+bnum", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["anum", "bnum", "bma", "atb", "knum", "apb"], "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{bnum}} \\frac{\\simplify{x-{bnum}}}{\\simplify{{knum}*x^2+{knum}*-1*{bma}x-{knum}*{atb}} }.\\]</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": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/({knum}*({anum}+{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"}, {"name": "Use algebra to evaluate limit 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "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": {}, "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"}, {"name": "Use algebra to evaluate limit 6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "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": {}, "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"}, {"name": "Use algebra to evaluate limit 7", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "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": {}, "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"}, {"name": "Use algebra to evaluate limit 8", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "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": {}, "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"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "extensions": [], "custom_part_types": [], "resources": []}