// Numbas version: exam_results_page_options
{"name": "Finding limits by substitution, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b1*a1^2+c1*a1+d1", "description": "", "name": "ans2"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b*a+c", "description": "", "name": "ans1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "c"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "d1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "a1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b2"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b3"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "c2"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "c1"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,-b3*a2,round(c2*b3/b2)])", "description": "", "name": "c3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "a"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "a2"}}, "ungrouped_variables": ["a", "c", "b", "ans1", "ans2", "a1", "a2", "b1", "b2", "b3", "c3", "c2", "c1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Finding limits by substitution, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans1", "minValue": "ans1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2", "minValue": "ans2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{b2*a2+c2}/{b3*a2+c3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Enter all numbers as either integers or fractions but not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>1. Find \\[\\lim_{x \\to \\var{a}}(\\simplify[std]{{b}x+{c}})\\]</p>\n        <p>Limit = [[0]].</p>\n        <p>2.Find \\[\\lim_{x \\to \\var{a1}}(\\simplify[std]{{b1}x^2+{c1}x+{d1}})\\]</p>\n        <p>Limit = [[1]].&nbsp;</p>\n        <p>3. Find \\[\\lim_{x \\to \\var{a2}}\\left(\\simplify[std]{({b2}x+{c2})/({b3}x+{c3})}\\right)\\]</p>\n        <p>Limit = [[2]]</p>\n        <p>Enter all numbers as either integers or fractions but not as decimals.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the following limits.</p>", "tags": ["checked2015", "limits", "MAS1601", "mas1601", "MAS1603"], "rulesets": {"std": ["all", "!noleadingMinus", "fractionNumbers", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Using simple substitution to find $\\lim_{x \\to a} bx+c$, $\\lim_{x \\to a} bx^2+cx+d$ and $\\displaystyle \\lim_{x \\to a} \\frac{bx+c}{dx+f}$ where $d\\times a+f \\neq 0$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>1. To find this limit we simply substitute $x=\\var{a}$ into $\\simplify[std]{{b}x+{c}}$ to get &nbsp;\\[\\lim_{x \\to \\var{a}}(\\simplify[std]{{b}x+{c}})=\\simplify[]{{b}*{a}+{c}={ans1}}\\]</p>\n    <p>2. Similarly to find this limit we simply substitute $x=\\var{a1}$ into $\\simplify[std]{{b1}x^2+{c1}x+{d1}}$ to get&nbsp; \\[\\lim_{x \\to \\var{a1}}(\\simplify[std]{{b1}x^2+{c1}x+{d1}})&nbsp;=\\simplify[]{{b1}*{a1}^2+{c1}*{a1}+{d1}={ans2}}\\]</p>\n    <p>3. Once again we could simply substitute $x=\\var{a2}$ into $\\displaystyle \\simplify[std]{({b2}x+{c2})/({b3}x+{c3})}$. However before doing this we must make sure that the denominator is not $0$ as otherwise we have a problem and the limit may not exist.</p>\n    <p>But $\\simplify[]{{b3}*{a2}+{c3}={b3*a2+c3} }\\neq 0$ and so we can make the substitution safely.</p>\n    <p>So &nbsp; \\[\\lim_{x \\to \\var{a2}}\\left(\\simplify[std]{({b2}x+{c2})/({b3}x+{c3})}\\right)=\\simplify[]{({b2}*{a2}+{c2})/({b3}*{a2}+{c3})}=\\simplify[all,fractionNumbers]{{b2*a2+c2}/{b3*a2+c3}}\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}