// Numbas version: exam_results_page_options
{"name": "limit 3 simple plug in type", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "limit 3 simple plug in type", "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "parts": [{"showCorrectAnswer": true, "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>", "type": "gapfill", "marks": 0, "scripts": {}, "gaps": [{"minValue": "ans1", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "allowFractions": false, "maxValue": "ans1", "correctAnswerFraction": false, "scripts": {}, "marks": 1}, {"minValue": "ans2", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "allowFractions": false, "maxValue": "ans2", "correctAnswerFraction": false, "scripts": {}, "marks": 1}, {"checkingaccuracy": 0.001, "vsetrange": [0, 1], "expectedvariablenames": [], "type": "jme", "checkingtype": "absdiff", "showpreview": true, "answer": "{b2*a2+c2}/{b3*a2+c3}", "checkvariablenames": false, "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "<p>Enter all numbers as either integers or fractions but not as decimals.</p>"}, "vsetrangepoints": 5, "marks": 1, "answersimplification": "std", "showCorrectAnswer": true, "scripts": {}}]}], "variables": {"b2": {"name": "b2", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c1": {"name": "c1", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c": {"name": "c", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ans1": {"name": "ans1", "definition": "b*a+c", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a1": {"name": "a1", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b": {"name": "b", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b3": {"name": "b3", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ans2": {"name": "ans2", "definition": "b1*a1^2+c1*a1+d1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "d1": {"name": "d1", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b1": {"name": "b1", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a2": {"name": "a2", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c2": {"name": "c2", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a": {"name": "a", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c3": {"name": "c3", "definition": "random(-9..9 except [0,-b3*a2,round(c2*b3/b2)])", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "variable_groups": [], "tags": ["checked2015", "limits", "MAS1601", "mas1601", "MAS1603"], "ungrouped_variables": ["a", "c", "b", "ans1", "ans2", "a1", "a2", "b1", "b2", "b3", "c3", "c2", "c1", "d1"], "type": "question", "rulesets": {"std": ["all", "!noleadingMinus", "fractionNumbers", "!collectNumbers"]}, "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>", "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>"}, "preamble": {"css": "", "js": ""}, "statement": "<p>Find the following limits.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}