Enter as a fraction or an integer and not as a decimal.
"}, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "showCorrectAnswer": true, "marks": 1, "checkvariablenames": false, "expectedvariablenames": [], "answersimplification": "all, fractionNumbers", "vsetrangepoints": 5}], "showCorrectAnswer": true, "prompt": "Find the following limit:
\n$\\displaystyle \\simplify{Limit(f(t),t,{d2}) }= \\;$[[0]].
\nEnter your answer as a fraction or an integer and not as a decimal.
", "scripts": {}, "marks": 0, "type": "gapfill"}, {"gaps": [{"answer": "({a3}*a+{b3})/({a2}*a^2+{2*a2*b2}*a+{c2+a2*b2^2})", "vsetrange": [0, 1], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "checkingaccuracy": 0.001, "answersimplification": "all", "showCorrectAnswer": true, "marks": 1, "checkvariablenames": false, "expectedvariablenames": [], "vsetrangepoints": 5}, {"choices": ["Because $f(t) \\neq 0, \\; \\forall t \\in \\mathbb{R}$.
", "Because $f$ is continuous at all points in $\\mathbb{R}$.
", "Because $f$ is a function defined in terms of polynomials.
", "Because all ratios of polynomials are continuous.
", "Because $f$ is differentiable at all points.
"], "displayColumns": 0, "distractors": ["Not true as $f(t)=0$ for a value of $t$.", "", "", "", ""], "scripts": {}, "maxMarks": 0, "matrix": [0, 1, 0, 0, 0], "minMarks": 0, "shuffleChoices": true, "showCorrectAnswer": true, "displayType": "radiogroup", "marks": 0, "type": "1_n_2"}], "showCorrectAnswer": true, "prompt": "Also find:
\n$\\displaystyle \\simplify{Limit(f(t),t,a) }= \\;$[[0]], where $a \\in \\mathbb{R}$ is any point.
\nWhy can we evaluate this limit? [[1]]
", "scripts": {}, "marks": 0, "type": "gapfill"}], "name": "Luis's copy of Find the limit of an algebraic fraction as parameter tends to a given value", "ungrouped_variables": ["w", "stat1", "b3", "a3", "a2", "b2", "v", "c2", "d2", "stat2"], "variables": {"c2": {"description": "", "definition": "random(1..5)", "templateType": "anything", "name": "c2", "group": "Ungrouped variables"}, "a3": {"description": "", "definition": "random(-9..9 except 0)", "templateType": "anything", "name": "a3", "group": "Ungrouped variables"}, "v": {"description": "", "definition": "a3*d2+b3", "templateType": "anything", "name": "v", "group": "Ungrouped variables"}, "w": {"description": "", "definition": "a2*d2^2+2*a2*b2*d2+c2+a2*b2^2", "templateType": "anything", "name": "w", "group": "Ungrouped variables"}, "b2": {"description": "", "definition": "-random(1..6)", "templateType": "anything", "name": "b2", "group": "Ungrouped variables"}, "d2": {"description": "", "definition": "random(0..5)", "templateType": "anything", "name": "d2", "group": "Ungrouped variables"}, "stat2": {"description": "", "definition": "(2*a2*b3-sqrt(4*a2^2*b3^2+4*a3*a2*(a3*c2-2*b3*b2*a2+a2*b2^2*a3)))/(-2*a3*a2)", "templateType": "anything", "name": "stat2", "group": "Ungrouped variables"}, "b3": {"description": "", "definition": "random(-9..9)", "templateType": "anything", "name": "b3", "group": "Ungrouped variables"}, "a2": {"description": "", "definition": "random(1..3)", "templateType": "anything", "name": "a2", "group": "Ungrouped variables"}, "stat1": {"description": "", "definition": "(2*a2*b3+sqrt(4*a2^2*b3^2+4*a3*a2*(a3*c2-2*b3*b2*a2+a2*b2^2*a3)))/(-2*a3*a2)", "templateType": "anything", "name": "stat1", "group": "Ungrouped variables"}}, "preamble": {"css": "", "js": ""}, "statement": "Let the function $f$ be given by $\\displaystyle f(t)=\\simplify{({a3} * t + {b3}) / ({a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2}) }$
", "metadata": {"description": "", "notes": "", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "functions": {"plotf": {"parameters": [["a2", "number"], ["b2", "number"], ["c2", "number"], ["a3", "number"], ["b3", "number"], ["stat1", "number"], ["stat2", "number"]], "definition": "var f = function(t){ return (a3*t+b3)/(a2*t*t+2*a2*b2*t+c2+a2*b2*b2); };\nvar m1=Math.min(stat1,stat2);\nvar m2=Math.max(stat1,stat2);\nvar f1=f(stat1);\nvar f2=f(stat2);\nvar a=Math.abs(f1);\nvar b=Math.abs(f2);\nvar M=Math.max(a,b);\nvar div = Numbas.extensions.jsxgraph.makeBoard('300px','300px', {axis:true,showNavigation:false,boundingbox:[m1-10,M+2,m2+10,-M-2]});\n\nvar brd=div.board;\n\nvar plot = brd.create('functiongraph',[f,m1-10,m2+10]);\n\n//brd.create('text',[c,-2,c]);\n//var i1 = brd.create('integral', [[0, c], plot]);\n\nreturn div;", "language": "javascript", "type": "html"}}, "variable_groups": [], "question_groups": [{"questions": [], "pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered"}], "showQuestionGroupNames": false, "advice": "Graph of $f$.
\n{plotf(a2,b2,c2,a3,b3,stat1,stat2)}
\n\nNote that $\\simplify{{a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2} ={a2}*(t+{b2})^2+{c2}} \\gt 0$.
\nHence the denominator of $f(t) \\neq 0,\\;\\forall t \\in \\mathbb{R}$ and so $f$ is continuous at all points in $\\mathbb{R}$.
\nThis means that in part a) we can take the limit by simply subsituting $t=\\var{d2}$ into the expression for $f(t)$ and we get:
\n\\[\\lim_{x \\to \\var{d2}}f(t)=\\simplify[all,!otherNumbers,fractionNumbers,!collectNumbers]{({a3} * {d2} + {b3}) / ({a2} * {d2}^ 2 + {2 * b2 * a2} * {d2}+ {c2 + a2 * b2 ^ 2})={v}/{w} }\\]
\nSimilarly in part b) we have :
\n\\[\\lim_{x \\to a}f(t)=\\simplify[all,!collectNumbers,!otherNumbers,fractionNumbers]{({a3} * a + {b3}) / ({a2} * a^ 2 + {2 * b2 * a2} * a+ {c2 + a2 * b2 ^ 2})}\\]
\n\nAs noted above we can find this limit by simply putting $t=a$ into the formula for the function as $f$ is continuous at all points in $\\mathbb{R}$.
\n", "type": "question", "tags": ["checked2015", "MAS2224"], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"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": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}