Let $\\displaystyle{a_n=\\var{a}^{1/n}}$
\n \n \n \n$\\displaystyle{\\lim_{n\\to \\infty} a_n=\\;\\;}$[[0]]
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \nLet $\\displaystyle{b_n=\\var{b}^{\\var{r}/n}}$
\n \n \n \n$\\displaystyle{\\lim_{n\\to \\infty} b_n=\\;\\;}$[[0]]
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "N", "maxValue": "N", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Consider the sequence $\\displaystyle{c_n=\\left(\\frac{1}{\\var{c}}\\right)^{\\var{s}/n}}$
\n1. Let $L=\\displaystyle{\\lim_{n\\to \\infty} c_n}$
\n$L=\\;\\;$[[0]]
\n2. Find the least positive integer $N$ such that
\\[|c_n-L| \\le 10^{-\\var{t}},\\;\\;\\forall n \\geq N\\]
Hint: You may assume that $c_n \\lt L,\\;\\;\\forall n$, hence $|c_n-L|=L-c_n,\\;\\;\\forall n$
\n$N=\\;\\;$[[1]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "\n \n \nWe have seen in the notes that $\\displaystyle{2^{1/n} \\longrightarrow 1}$ as $n \\longrightarrow \\infty$.
\n \n \n \nAnswer the following questions:
\n \n ", "tags": ["checked2015", "finding the limit of a sequence", "limit", "limits", "MAS1601", "sequence", "sequences", "taking the limit", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/07/2012:
\nAdded tags.
\nAdded explanation in Advice why inequality remains as $\\le$ on taking $\\ln$ of both sides.
\n22/07/2012:
\nAdded description.
\nChanged $n \\ge N$ to $n \\geq N$ as otherwise $N-1$ would be be the correct answer not $N$.
\nFirst two parts are trivial.
\n27/7/2012:
\nAdded tags.
\nIn the Advice section, moved \\Rightarrow to the beginning of the line instead of the end of the previous line.
\nQuestion appears to be working correctly.
\n25/12/2012:
\nChecked calculation, OK. Added tested1 tag.
\nChanged all $\\lt$ etc to $\\le$. Also edited the description for this.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Question on $\\displaystyle{\\lim_{n\\to \\infty} a^{1/n}=1}$. Find least integer $N$ s.t. $\\ \\left |1-\\left(\\frac{1}{c}\\right)^{b/n}\\right| \\le10^{-r},\\;n \\geq N$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nIt is true that for any positive real number $a$ that $\\displaystyle{\\lim_{n\\to \\infty} a^{1/n}=1}$
\nHence $\\displaystyle{\\lim_{n\\to \\infty} \\var{a}^{1/n}=1}$
\nb)
We have $\\displaystyle{b_n=\\var{b}^{\\var{r}/n}=\\left(\\var{b}^{\\var{r}}\\right)^{1/n}}$
Hence by the result quoted in part a) we have $\\displaystyle{\\lim_{n\\to \\infty} b_n=1}$
\nc)
\n1. Once again by the result quoted in a) and writing $\\displaystyle{\\left(\\frac{1}{\\var{c}} \\right)^{\\var{s}/n}=\\left(\\frac{1}{\\var{c}^{\\var{s}}} \\right)^{1/n}}$
\nwe have:
\n\\[L=\\lim_{n\\to \\infty} c_n = \\lim_{n \\to \\infty}\\left(\\frac{1}{\\var{c}^{\\var{s}}} \\right)^{1/n} = 1\\]
\n2. In order to find the least $N$ such that $\\displaystyle{|c_n-1| \\le 10^{-\\var{t}},\\;\\;\\forall n \\ge N}$ we write this using the hint as:
\n\\[\\begin{eqnarray*} 1-\\left(\\frac{1}{\\var{c}}\\right)^{\\var{s}/n} &\\le& \\frac{1}{10^{\\var{t}}} \\\\ \\Rightarrow \\left(\\frac{1}{\\var{c}}\\right)^{\\var{s}/n} &\\ge& 1-\\frac{1}{10^{\\var{t}}}=\\frac{\\var{10^t-1}}{\\var{10^t}} \\\\ \\Rightarrow \\var{c}^{\\var{s}/n} &\\le& \\frac{\\var{10^t}}{\\var{10^t-1}} \\end{eqnarray*} \\]
\nTaking $\\ln$ of both sides gives:(the inequality sign remains as it is as $\\ln(\\var{c}) \\gt 0 $ and $\\displaystyle \\ln\\left(\\frac{\\var{10^t}}{\\var{10^t-1}}\\right) \\gt 0$ )
\n\\[\\begin{eqnarray*} \\frac{\\var{s}}{n}\\ln(\\var{c}) &\\le& \\ln\\left(\\frac{\\var{10^t}}{\\var{10^t-1}}\\right)\\\\ \\Rightarrow n &\\ge& \\frac{\\var{s}\\ln(\\var{c})}{ \\ln\\left(\\frac{\\var{10^t}}{\\var{10^t-1}}\\right)}=\\var{trN} \\end{eqnarray*} \\]
\nOn rounding up the least $N$ is $N=\\var{N}$.
"}, {"name": "Find limit of a sequence", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["a", "b1", "b", "c", "d"], "name": "x_n"}, {"variables": ["n1", "n2", "n3", "n4"], "name": "Wrong choices for N"}], "variables": {"b": {"templateType": "anything", "group": "x_n", "definition": "if(a*d=b1*c,b1+1,b1)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "x_n", "definition": "random(1..20)", "description": "", "name": "c"}, "n1": {"templateType": "anything", "group": "Wrong choices for N", "definition": "N+\nif(\n N>50,\n random(-1,1)*random(8..48#4),\n //otherwise\n random(1,3)\n)", "description": "", "name": "n1"}, "b1": {"templateType": "anything", "group": "x_n", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "b1"}, "a": {"templateType": "anything", "group": "x_n", "definition": "random(2..9)", "description": "", "name": "a"}, "n3": {"templateType": "anything", "group": "Wrong choices for N", "definition": "N+\nif(\n N>50,\n random(-1,1)*random(6..42#4),\n //otherwise\n random(5,6)\n)", "description": "", "name": "n3"}, "n2": {"templateType": "anything", "group": "Wrong choices for N", "definition": "N+\nif(\n N>50,\n random(-1,1)*random(7..43#4),\n //otherwise\n random(2,4)\n)", "description": "", "name": "n2"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,4)", "description": "Power of 10 to get within.
", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(tval)", "description": "Smallest $N$ such that $x_n$ is within ep of its limit.
Value of $n$ when $x_n$ is exactly $10^{-r}$ away from the limit (not necessarily an integer)
", "name": "tval"}, "d": {"templateType": "anything", "group": "x_n", "definition": "random(1..20)", "description": "", "name": "d"}, "ep": {"templateType": "anything", "group": "Ungrouped variables", "definition": "10^(-r)", "description": "", "name": "ep"}}, "ungrouped_variables": ["r", "ep", "tval", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a}/{c}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Enter your answer as a fraction or integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n \n \nWhat is the limit of this sequence?
\n \n \n \n$\\displaystyle{\\lim_{x\\to\\infty} x_n=\\;\\;}$[[0]]
\n \n \n \nInput the limit as a fraction or an integer and not a decimal.
\n \n \n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{N}
", "{N1}
", "{N2}
", "{N3}
", "{N4}
"], "displayColumns": 0, "distractors": ["", "", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0, 0], "marks": 0}], "type": "gapfill", "prompt": "\n \n \nWhich of the following integers has the property that it is the least integer $N$ such that all terms in the sequence are within $10^{\\var{-r}}$ of the limit for all $n \\geq N $?
\n \n \n \n[[0]]
\n \n \n", "showCorrectAnswer": true, "marks": 0}], "statement": "Let
\n\\[ x_n=\\simplify[std]{({a}n+{b})/({c}n+{d})}, \\quad n=1,\\; 2,\\; 3 \\ldots \\]
", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of sequences", "limits", "MAS1601", "sequences", "taking the limit", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "4/07/2012:
\n
Changed inequality sign in prompt from $\\lt$ to $\\leq$ and as a consequence changed them in the Advice. Answer remains the same.
21/07/2012:
\nAdded description.
\nNeeds better tags to describe second part.
\nAlso need to redefine the variables so that a and b and a and c are coprime - results in a better and less clumsy Advice solution. This is the \"changes needed\" tag. Issue raised as having defined a new function chcop using the gcd function, the editor did not register it in the variables list - although the question compiled and ran.
\n(Contd.) The variables a,b,c,d have been redefined. Also noticed that the MCQ had two correct answers on rare occasions. This has been corrected.
\nGot rid of the changes needed tag.
\n27/7/2012:
\nAdded tags.
\n24/12/2012:
\nChecked calculation, OK. Added tested1 tag.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Let $x_n=\\frac{an+b}{cn+d},\\;\\;n=1,\\;2\\ldots$. Find $\\lim_{x \\to\\infty} x_n=L$ and find least $N$ such that $|x_n-L| \\le 10^{-r},\\;n \\geq N,\\;r \\in \\{2,\\;3,\\;4\\}$.
"}, "variablesTest": {"condition": "gcd(a,b)=1 and gcd(c,d)=1", "maxRuns": 100}, "advice": "The limit is $\\displaystyle \\simplify[std]{{a}/{c}}$.
\nTo find the least $N$ such that all terms from the the $N$th are within $10^{\\var{-r}}$ of the limit, we proceed as follows:
\n\\begin{align}
\\left| \\simplify[std]{x_n -({a} / {c})} \\right| \\leq 10^{ -\\var{r}} &\\iff \\left| \\simplify[std]{({a}n+{b})/({c}n+{d})-{a}/{c}} \\right| \\leq 10 ^ { -\\var{r}} \\\\
&\\iff \\simplify[std]{abs({b*c-a*d})/({c^2}n+{c*d})}\\leq 10 ^ { -\\var{r}}
\\end{align}
(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n+{c*d}} \\gt 0$, $\\forall n=1,\\; 2,\\; 3 \\ldots $)
\nRearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n+{c*d}}) \\times 10^{\\var{r}}$ (this is positive and so the inequality does not reverse), we get:
\n\\[ \\simplify[std]{{c^2}n+{c*d}} \\geq \\var{10^r*abs(b*c-a*d)} \\iff n \\geq \\simplify[std]{{1}/{c^2}({10^r*abs(b*c-a*d)}-{c*d})}=\\var{tval}\\]
\n{if(fract(tval)>0,\"The least integer value is given by rounding up, i.e.\",\"So\")} $N=\\var{N}$.
"}, {"name": "Find limits of sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["r", "v15", "v110", "v150"], "name": "Part a"}, {"variables": ["k1", "v25", "v210", "v250"], "name": "Part b"}, {"variables": ["k", "v35", "v310", "v350"], "name": "Part c"}, {"variables": ["c", "d", "al", "ga", "v45", "v410", "v450"], "name": "Part d"}, {"variables": ["n", "v55", "v510", "v550", "v560"], "name": "Part e"}, {"variables": ["a3", "ab", "b3", "v65", "v610", "v650", "v660", "val", "tol", "valexp"], "name": "Part f"}, {"variables": ["a", "b"], "name": "Part g"}], "variables": {"v150": {"templateType": "anything", "group": "Part a", "definition": "precround(5000000^(-1/r),5)", "description": "", "name": "v150"}, "v210": {"templateType": "anything", "group": "Part b", "definition": "precround(k1^(1/5000),5)", "description": "", "name": "v210"}, "b3": {"templateType": "anything", "group": "Part f", "definition": "switch(ab=1,random(2..9),ab=2,random(3,5,7,9),ab=3,random(2,4,5,7,8),random(3,5,7,9))", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s5"}, "ab": {"templateType": "anything", "group": "Part f", "definition": "abs(a3)", "description": "", "name": "ab"}, "v65": {"templateType": "anything", "group": "Part f", "definition": "precround((1 + s5 * (abs(a3) / (b3 * 10))) ^ 10,5)", "description": "", "name": "v65"}, "v560": {"templateType": "anything", "group": "Part e", "definition": "precround((c/n)^89,5)", "description": "", "name": "v560"}, "v650": {"templateType": "anything", "group": "Part f", "definition": "precround((1 + s5 * (abs(a3) / (b3 * 1000))) ^ 1000,5)", "description": "", "name": "v650"}, "k": {"templateType": "anything", "group": "Part c", "definition": "random(2..20#0.5)", "description": "", "name": "k"}, "k1": {"templateType": "anything", "group": "Part b", "definition": "random(100000..200000)", "description": "", "name": "k1"}, "v550": {"templateType": "anything", "group": "Part e", "definition": "precround((c/n)^50,5)", "description": "", "name": "v550"}, "b": {"templateType": "anything", "group": "Part g", "definition": "a*d-random(1..3)", "description": "", "name": "b"}, "v510": {"templateType": "anything", "group": "Part e", "definition": "precround((c/n)^29,5)", "description": "", "name": "v510"}, "v450": {"templateType": "anything", "group": "Part d", "definition": "precround((c * 5000000 + d) / (al * 5000000 -ga),5)", "description": "", "name": "v450"}, "v410": {"templateType": "anything", "group": "Part d", "definition": "precround((c * 5000 + d) / (al * 5000 -ga),5)", "description": "", "name": "v410"}, "v35": {"templateType": "anything", "group": "Part c", "definition": "precround(k^(1/100),5)", "description": "", "name": "v35"}, "d": {"templateType": "anything", "group": "Part d", "definition": "random(2..5)", "description": "", "name": "d"}, "a3": {"templateType": "anything", "group": "Part f", "definition": "s5*random(1..4)", "description": "", "name": "a3"}, "ga": {"templateType": "anything", "group": "Part d", "definition": "random(2..5)", "description": "", "name": "ga"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(d2=d,d+1,d2)", "description": "", "name": "d1"}, "r": {"templateType": "anything", "group": "Part a", "definition": "random(2..10)", "description": "", "name": "r"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "c1"}, "v110": {"templateType": "anything", "group": "Part a", "definition": "precround(5000^(-1/r),5)", "description": "", "name": "v110"}, "v350": {"templateType": "anything", "group": "Part c", "definition": "precround(k^(1/5000000),5)", "description": "", "name": "v350"}, "v660": {"templateType": "anything", "group": "Part f", "definition": "precround((1 + s5 * (abs(a3) / (b3 * 10000))) ^ 10000,5)", "description": "", "name": "v660"}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "tol": {"templateType": "anything", "group": "Part f", "definition": "0", "description": "", "name": "tol"}, "v250": {"templateType": "anything", "group": "Part b", "definition": "precround(k1^(1/5000000),5)", "description": "", "name": "v250"}, "a": {"templateType": "anything", "group": "Part g", "definition": "random(2..4)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Part d", "definition": "s1*random(11..50)", "description": "", "name": "c"}, "valexp": {"templateType": "anything", "group": "Part f", "definition": "precround(exp(a3/b3),5)", "description": "", "name": "valexp"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "d2"}, "val": {"templateType": "anything", "group": "Part f", "definition": "precround(exp(a3/b3),4)", "description": "", "name": "val"}, "n": {"templateType": "anything", "group": "Part e", "definition": "abs(c)+random(2..5)", "description": "", "name": "n"}, "de": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "name": "de"}, "be": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "name": "be"}, "v15": {"templateType": "anything", "group": "Part a", "definition": "precround(100^(-1/r),5)", "description": "", "name": "v15"}, "v610": {"templateType": "anything", "group": "Part f", "definition": "precround((1 + s5 * (abs(a3) / (b3 * 100))) ^ 100,5)", "description": "", "name": "v610"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*d1-1", "description": "", "name": "b1"}, "al": {"templateType": "anything", "group": "Part d", "definition": "random(1..4)", "description": "", "name": "al"}, "v55": {"templateType": "anything", "group": "Part e", "definition": "precround((c/n)^10,5)", "description": "", "name": "v55"}, "v310": {"templateType": "anything", "group": "Part c", "definition": "precround(k^(1/5000),5)", "description": "", "name": "v310"}, "v25": {"templateType": "anything", "group": "Part b", "definition": "precround(k1^(1/100),5)", "description": "", "name": "v25"}, "v45": {"templateType": "anything", "group": "Part d", "definition": "precround((c * 100 + d) / (al * 100 -ga),5)", "description": "", "name": "v45"}}, "ungrouped_variables": ["b1", "be", "c1", "d1", "d2", "de", "s1", "s5"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "0", "minValue": "0", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^{1/\\var{r}}}\\right)=} $ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\var{k1}^{1/n}\\right)=} $ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\var{k}^{1/n}\\right)=} $ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "scripts": {}, "type": "numberentry", "maxValue": "c/al", "minValue": "c/al", "correctAnswerFraction": true, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$\\displaystyle \\lim_{n \\to \\infty}\\left(\\simplify[std]{({c}n+{d})/({al}n-{ga})}\\right) = $ [[0]]
\nEnter your answer as a fraction or integer, not as a decimal.
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "0", "minValue": "0", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify[std]{{c}/{n}}\\right)^n=} $ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "e^({a3}/{b3})", "vsetrange": [0, 1], "checkingaccuracy": "0.0001", "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify[std]{1+{a3}/({b3}n)}\\right)^n=} $ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "scripts": {}, "type": "numberentry", "maxValue": "al/ga", "minValue": "al/ga", "correctAnswerFraction": true, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$\\displaystyle{\\lim_{n \\to \\infty}\\frac{\\left(\\simplify[std]{{al^d}n^({a*d})+{be}n^{b}+{c}}\\right)^{1/\\var{d}}} {\\left(\\simplify[std]{{ga^d1}n^({a*d1})+{de}n^{b1}+{c1}}\\right)^{1/\\var{d1}}}=} $ [[0]]
\nEnter your answer as a fraction or integer, not as a decimal.
", "showCorrectAnswer": true, "marks": 0}], "statement": "What are the following limits?
", "tags": ["checked2015", "examples of standard limits", "limit", "limits", "limits of sequences", "MAS1601", "sequences", "standard limits", "taking the limit"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t4/07/2012:
\n\t\tAdded tags.
\n\t\tImproved display of prompt for fourth part.
\n\t\tImproved display of solution to fourth part.
\n\t\tChecked calculations.
\n\t\tNo tolerance on answer to 6th part, got to be exact to 4dps. Tolerance variable, tol=0.
\n\t\t21/07/2012:
\n\t\tAdded description.
\n\t\t27/7/2012:
\n\t\tAdded tags.
\n\t\tQuestion appears to be working correctly.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Seven standard elementary limits of sequences.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "All calculations below are rounded to $5$ decimal places.
\nThe notation $a \\approx b$ means that $a$ and $b$ are approximately equal.
\nUsing a calculator for $3$ values of $n$:
\n| $n$ | $\\displaystyle{\\frac{1}{n^{1/\\var{r}}}}$ |
|---|---|
| $100$ | \n$\\var{v15}$ | \n
| $5000$ | \n$\\var{v110}$ | \n
| $5000000$ | \n$\\var{v150}$ | \n
This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^{1/\\var{r}}}\\right)=0}$
\nIn fact $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^r}\\right)=0}$ for any $r \\gt 0$
\n| $n$ | $\\displaystyle{\\var{k1}^{1/n}}$ |
|---|---|
| $100$ | \n$\\var{v25}$ | \n
| $5000$ | \n$\\var{v210}$ | \n
| $5000000$ | \n$\\var{v250}$ | \n
This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\var{k1}^{1/n}=1}$.
\n| $n$ | $\\displaystyle{\\var{k}^{1/n}}$ |
|---|---|
| $100$ | \n$\\var{v35}$ | \n
| $5000$ | \n$\\var{v310}$ | \n
| $5000000$ | \n$\\var{v350}$ | \n
This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\var{k}^{1/n}=1}$.
\nFrom the last two questions it seems that $\\displaystyle{\\lim_{n \\to \\infty} k^{1/n}=1}$ for any $k \\gt 0$ – and this is in fact true.
\n| $n$ | $\\displaystyle{\\frac{\\var{c}n+\\var{d}}{\\var{al}n-\\var{ga}}}$ |
|---|---|
| $100$ | \n$\\var{v45}$ | \n
| $5000$ | \n$\\var{v410}$ | \n
| $5000000$ | \n$\\var{v450}$ | \n
This indicates that $\\displaystyle \\lim_{n \\to \\infty}\\left(\\simplify[std]{({c}n+{d})/({al}n-{ga})}\\right) = \\simplify[std]{{c}/{al}}$.
\nIn general, $\\displaystyle{ \\lim_{n \\to \\infty}\\left(\\frac{an+b}{cn+d}\\right)= \\frac{a}{c} }$ when $c \\neq 0$.
\n| $n$ | $\\displaystyle{\\left(\\simplify{{c}/{n}}\\right)^n}$ |
|---|---|
| $10$ | \n$\\var{v55}$ | \n
| $29$ | \n$\\var{v510}$ | \n
| $50$ | \n$\\var{v550}$ | \n
| $89$ | \n$\\var{v560}$ | \n
This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify{{c}/{n}}\\right)^n}= 0$.
\nIn general $\\displaystyle{\\lim_{n \\to \\infty} r^n= 0}$ if $|r| \\lt 1$.
\nWe have the limit:
\n\\[ \\lim_{n\\to\\infty}\\left(1+\\frac{a}{n}\\right) = e^a \\]
\nThe following table confirms that the values are converging to $\\displaystyle{\\simplify[std]{e^({a3}/{b3})={valexp}}}$ (to five decimal places).
\n| $n$ | $\\displaystyle{\\left(\\simplify[std]{1+{a3}/({b3}n)}\\right)^n}$ |
|---|---|
| $10$ | \n$\\var{v65}$ | \n
| $100$ | \n$\\var{v610}$ | \n
| $1000$ | \n$\\var{v650}$ | \n
| $10000$ | \n$\\var{v660}$ | \n
Hence the answer asked for is $\\var{val}$ to $4$ decimal places.
\nThe answer to this question is based upon neglecting terms in polynomials in $n$ for large $n$.
\nFor example, $n^3+1000000n^2+1000000000 \\approx n^3$ for large $n$ because the $n^3$ term completely dominates the other terms as $n \\to \\infty$.
\nA more precise way of saying this is:
\n\\[\\lim_{n\\to\\infty}\\left(\\frac{n^3+1000000n^2+1000000000}{n^3}\\right)=1\\]
\nSo for large $n$,
\n\\begin{align}
\\frac{\\left(\\simplify[std]{{al^d}n^({a*d})+{be}n^{b}+{c}}\\right)^{1/\\var{d}}} {\\left(\\simplify[std]{{ga^d1}n^({a*d1})+{de}n^{b1}+{c1}}\\right)^{1/\\var{d1}}} &\\approx \\frac{\\left(\\simplify[std]{{al^d}n^({a*d})}\\right)^{1/\\var{d}}} {\\left(\\simplify[std]{{ga^d1}n^({a*d1})}\\right)^{1/\\var{d1}}} \\\\ \\\\
&= \\frac{\\simplify[std]{{al^d}^(1/{d})n^{a}}} {\\simplify[std]{{ga^d1}^(1/{d1})n^{a}}} \\\\ \\\\
&= \\simplify[std]{{al}/{ga}}
\\end{align}
Hence the limit is $\\displaystyle{\\simplify[std]{{al}/{ga}}}$
"}, {"name": "Finding limits by substitution, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "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}], "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": "Enter all numbers as either integers or fractions but not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "1. Find \\[\\lim_{x \\to \\var{a}}(\\simplify[std]{{b}x+{c}})\\]
\nLimit = [[0]].
\n2.Find \\[\\lim_{x \\to \\var{a1}}(\\simplify[std]{{b1}x^2+{c1}x+{d1}})\\]
\nLimit = [[1]].
\n3. Find \\[\\lim_{x \\to \\var{a2}}\\left(\\simplify[std]{({b2}x+{c2})/({b3}x+{c3})}\\right)\\]
\nLimit = [[2]]
\nEnter all numbers as either integers or fractions but not as decimals.
", "showCorrectAnswer": true, "marks": 0}], "statement": "Find the following limits.
", "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": "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 $d\\times a+f \\neq 0$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "1. To find this limit we simply substitute $x=\\var{a}$ into $\\simplify[std]{{b}x+{c}}$ to get \\[\\lim_{x \\to \\var{a}}(\\simplify[std]{{b}x+{c}})=\\simplify[]{{b}*{a}+{c}={ans1}}\\]
\n2. Similarly to find this limit we simply substitute $x=\\var{a1}$ into $\\simplify[std]{{b1}x^2+{c1}x+{d1}}$ to get \\[\\lim_{x \\to \\var{a1}}(\\simplify[std]{{b1}x^2+{c1}x+{d1}}) =\\simplify[]{{b1}*{a1}^2+{c1}*{a1}+{d1}={ans2}}\\]
\n3. 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.
\nBut $\\simplify[]{{b3}*{a2}+{c3}={b3*a2+c3} }\\neq 0$ and so we can make the substitution safely.
\nSo \\[\\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}}\\]
"}, {"name": "True/false statements about limits of sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"displayType": "radiogroup", "layout": {"type": "all", "expression": ""}, "choices": ["{statements[choices[0]]}", "{statements[choices[1]]}", "{statements[choices[2]]}", "{statements[choices[3]]}", "{statements[choices[4]]}", "{statements[choices[5]]}"], "showCorrectAnswer": true, "matrix": "map(statement_marks[choices[j]],j,0..5)", "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": true, "answers": ["True", "False"], "scripts": {}, "maxMarks": 0, "minAnswers": 0, "minMarks": 0, "shuffleAnswers": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "warningType": "none"}], "type": "gapfill", "prompt": "\n \n \n[[0]]
\n \n \n ", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variables": {"t_statements": {"templateType": "list of strings", "group": "Statements", "definition": "[ \"A convergent sequence is bounded.\", \"An unbounded sequence does not converge.\", \"It is possible for a sequence to be both increasing and decreasing.\", \"A sequence with only a finite number of non zero terms converges to 0.\", \"A bounded increasing sequence converges.\", \"A bounded decreasing sequence converges.\", \"There are convergent sequences with all terms greater than zero and the limit is 0.\" ]", "name": "t_statements", "description": ""}, "f_statements": {"templateType": "list of strings", "group": "Statements", "definition": "[ \"A bounded sequence is convergent.\", \"Every divergent sequence is unbounded.\", \"Every convergent sequence is monotone.\", \"There are convergent sequences with all terms greater than zero and the limit is less than zero.\", \"There are bounded monotone sequences that do not converge.\", \"All convergent sequences of positive terms converge to a value greater than zero.\", \"There are unbounded sequences which converge.\", \"It is not possible for a sequence to be both increasing and decreasing.\" ]", "name": "f_statements", "description": ""}, "choices": {"templateType": "anything", "group": "Statements", "definition": "shuffle(0..len(statements)-1)[0..6]", "name": "choices", "description": ""}, "statements": {"templateType": "anything", "group": "Statements", "definition": "t_statements+f_statements", "name": "statements", "description": ""}, "statement_marks": {"templateType": "anything", "group": "Statements", "definition": "map([1,-1],x,t_statements)+map([-1,1],x,f_statements)", "name": "statement_marks", "description": ""}}, "ungrouped_variables": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [{"variables": ["t_statements", "f_statements", "statements", "statement_marks", "choices"], "name": "Statements"}], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n \n \nAnswer the following question on limits of sequences. Note that a sequence is unbounded if it is not bounded.
\n \n \n \nNote that every correct answer is worth 1 mark – but every wrong answer loses a mark.
\n \n \n ", "tags": ["bounded sequences", "checked2015", "convergence", "convergent sequences", "decreasing sequence", "divergence", "divergent sequences", "increasing sequence", "limits", "mas1601", "MAS1601", "monotone sequence", "monotonic sequence", "sequence", "sequences", "subsequence", "tested1", "unbounded sequences"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/07/2012:
\nAdded tags.
\n21/07/2012:
\nAdded description.
\n27/7/2012:
\nAdded tags.
\n24/12/2012:
\nChecked choices, OK. Added tested1 tag.
\nAlso included escaped slashes in string variables including tex.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (3 correct out of 6) re properties of convergent and divergent sequences. Selection of questions from a pool.
"}, "advice": "\n \n \nCheck against your course notes.
\n \n \n "}, {"name": "True/false statements about limits of sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["t_statements", "f_statements", "statements", "statement_marks", "choices"], "name": "Statements"}], "variables": {"t_statements": {"group": "Statements", "templateType": "list of strings", "definition": "[ \"There exists a sequence with all terms greater than zero and with limit 0.\", \"A divergent sequence can have a convergent subsequence.\", \"A convergent sequence is bounded.\", \"If a sequence is not bounded, then it does not converge.\", \"If {$\\{x_{n+i}\\}$} diverges for some natural number $i$, then {$\\{x_n\\}$} diverges.\", \"It is possible for a sequence to be both increasing and decreasing.\", \"In a convergent sequence, all subsequences converge to the same limit.\", \"A bounded increasing sequence converges.\", \"A bounded decreasing sequence converges.\", \"A sequence with only a finite number of non zero terms converges to 0.\", \"If a sequence has the subsequence given by the even terms converges to the same limit as the subsequence of odd terms, then the sequence also converges to that limit.\", \"A set with a maximum element is necessarily bounded above.\", \"A set with a minimum element is necessarily bounded below.\", \"A finite set is bounded.\", \"A set that is not bounded has an infinite number of elements.\", \"A bounded set has both a least upper bound and a greatest lower bound.\" ]", "description": "", "name": "t_statements"}, "f_statements": {"group": "Statements", "templateType": "list of strings", "definition": "[ \"A bounded sequence is convergent.\", \"Every divergent sequence is unbounded.\", \"A convergent sequence is either increasing or decreasing.\", \"If {$\\{x_n\\}$} converges, then {$\\{x_{n+i}\\}$} could diverge for some natural number $i$.\", \"There exists a convergent sequence {$\\{x_n\\}$} with $x_n >0$ for all $n \\\\in \\\\mathbb{N}$ and limit $\\\\ell <0$.\", \"There exists a bounded increasing sequences that does not converge.\", \"All convergent sequences of positive terms converge to a value $> 0$.\", \"If {$\\{x_n\\}$} diverges, then {$\\{x_{n+i}\\}$} could converge for some natural number $i$.\", \"There exists a sequence that is not bounded but which converges.\", \"It is not possible for a sequence to be both increasing and decreasing.\", \"A set with a maximum element is necessarily bounded.\", \"A set with a minimum element is necessarily bounded.\", \"A bounded set necessarily has a maximum element.\", \"A bounded set necessarily has a minimum element.\", \"A bounded set never has a maximum element.\", \"A bounded set never has a minimum element.\", \"A set with an infinite number of elements cannot be bounded.\" ]", "description": "", "name": "f_statements"}, "statement_marks": {"group": "Statements", "templateType": "anything", "definition": "map([1,-1],x,t_statements)+map([-1,1],x,f_statements)", "description": "", "name": "statement_marks"}, "statements": {"group": "Statements", "templateType": "anything", "definition": "t_statements+f_statements", "description": "", "name": "statements"}, "choices": {"group": "Statements", "templateType": "anything", "definition": "shuffle(0..len(statements)-1)[0..8]", "description": "", "name": "choices"}}, "ungrouped_variables": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"prompt": "\n \n \n[[0]]
\n \n \n \n", "scripts": {}, "gaps": [{"displayType": "radiogroup", "minMarks": 0, "layout": {"type": "all", "expression": ""}, "choices": ["{statements[choices[0]]}", "{statements[choices[1]]}", "{statements[choices[2]]}", "{statements[choices[3]]}", "{statements[choices[4]]}", "{statements[choices[5]]}", "{statements[choices[6]]}", "{statements[choices[7]]}"], "variableReplacementStrategy": "originalfirst", "matrix": "map(statement_marks[x],x,choices)", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "warningType": "none", "scripts": {}, "marks": 0, "type": "m_n_x", "maxMarks": 0, "shuffleAnswers": false, "showCorrectAnswer": true, "variableReplacements": [], "answers": ["True", "False"]}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Answer the following question on sequences and sets. Note that a sequence is said to be unbounded if it is not bounded.
\nNote that every correct answer is worth 1 mark, but every wrong answer loses a mark.
", "tags": ["bounded sequences", "bounded sets", "checked2015", "convergence", "convergent sequences", "decreasing sequence", "divergence", "divergent sequences", "increasing sequence", "limits", "MAS1601", "monotone sequence", "monotonic sequence", "sequence", "sequences", "subsequence", "unbounded sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/07/2012:
\nAdded tags.
\n21/07/2012:
\nAdded description.
\n27/7/2012:
\nAdded tags.
\n24/12/2012:
\nChecked choices, OK. Added tested1 tag.
\nAlso included escaped slashes in string variables including tex.
\n28/01/2015:
\n(OK) Adjusted questions on sequences and added questions on boundedness of sets.
\n17/04/2015:
\n(OK) Recast the statements as long text.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (4 correct out of 8) covering properties of convergent and divergent sequences and boundedness of sets. Selection of questions from a pool.
"}, "advice": "You should be able to work out the correct answers from your notes.
"}, {"name": "True/false statements about properties of continuity and limits, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "v"}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If for some sequence {$u_n$} converging to $c$, the sequence {$f(u_n)$} converges to $l$, then $f(x) \\\\to l$ as $x \\\\to c$.
\"", "description": "", "name": "f1"}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))", "description": "", "name": "ch2"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "u"}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f(x) \\\\not\\\\to l$ as $x$ tends to $c$, then $f(x_n) \\\\not\\\\to l$ as $n \\\\to \\\\infty$ for every sequence {$x_n$} converging to $c$.
\"", "description": "", "name": "f4"}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If the limit of $f(x)$ as $x \\\\to c$ exists then the limit is $f(c)$.
\"", "description": "", "name": "f5"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "h"}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If for some sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$, then the function $f$ is continuous at $c$.
\"", "description": "", "name": "f2"}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f(x) \\\\to \\\\ell$ as $x \\\\to c$ and if $g(x) \\\\to m$ as $x \\\\to c$, then $\\\\dfrac{f(x)}{g(x)} \\\\to \\\\dfrac{\\\\ell}{m}$ as $x \\\\to c$.
\"", "description": "", "name": "f3"}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"There exists a function $f$ such that the limit of $f(x)$ as $x \\\\to c$ exists and $f(c)$ exists, but $f$ is not continuous at $c$.
\"", "description": "", "name": "tr5"}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))", "description": "", "name": "ch1"}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If the limit of $f(x)$ as $x \\\\to c$ exists and the limit is $f(c)$, then $f$ is continuous at $c$.
\"", "description": "", "name": "tr6"}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous at $c$, then for any sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$.
\"", "description": "", "name": "tr2"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "f"}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If for every sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$, then the function $f$ is continuous at $c$.
\"", "description": "", "name": "tr3"}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If the limit of $f(x)$ as $x \\\\to c$ exists and if $f(c)$ exists, then $f$ is continuous at $c$.
\"", "description": "", "name": "f6"}, "ch3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,f1,if(v=2,f2,f3))", "description": "", "name": "ch3"}, "ch4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f4,if(f=2,f5,f6))", "description": "", "name": "ch4"}, "tr4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f(x) \\\\not\\\\to \\\\ell$ as $x \\\\to c$, then $f(x_n) \\\\not\\\\to \\\\ell$ as $n \\\\to \\\\infty$ for some sequence {$x_n$} converging to $c$.
\"", "description": "", "name": "tr4"}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f(x) \\\\to \\\\ell$ as $x \\\\to c$ and if $x_n \\\\to c$ as $n \\\\to \\\\infty$ (with each $x_n \\\\neq c$), then $f(x_n) \\\\to \\\\ell$ as $n \\\\to \\\\infty$.
\"", "description": "", "name": "tr1"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "g"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "t"}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "t", "tr1", "u", "tr2", "tr3", "tr4", "tr5", "tr6", "g", "f", "h", "ch1", "ch2", "ch3", "ch4", "v"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n \n \n[[0]]
\n \n \n \n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"displayType": "checkbox", "showCellAnswerState": true, "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minAnswers": 0, "minMarks": 0, "shuffleAnswers": false, "variableReplacements": [], "layout": {"expression": ""}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": [[1, -1], [1, -1], ["-1", "1"], [-1, 1]], "choices": ["{Ch1}
", "{Ch2}
", "{Ch3}
", "{Ch4}
"], "unitTests": [], "marks": 0, "scripts": {}, "maxMarks": 0, "showCorrectAnswer": true, "answers": ["True", "False"], "warningType": "none"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Answer the following question on continuity and limits of functions. You may assume that the functions $f$ are $:\\mathbb{R} \\to \\mathbb{R}$. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.
", "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (2 correct out of 4) covering properties of continuity and limits of functions. Selection of questions from a pool.
"}, "advice": "You should be able to work out the correct answers from your notes.
"}, {"name": "When is a sequence within d of its limit?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["a", "b1", "b", "c", "d"], "name": "x_n"}, {"variables": ["r1", "ep1", "tval", "N_1"], "name": "Part a"}, {"variables": ["r2", "ep2", "N_2"], "name": "Part b"}], "variables": {"ep1": {"templateType": "anything", "group": "Part a", "definition": "10^(-r1)", "name": "ep1", "description": ""}, "r1": {"templateType": "anything", "group": "Part a", "definition": "random(2,3,4,5,6)", "name": "r1", "description": ""}, "d": {"templateType": "anything", "group": "x_n", "definition": "random(1..20)", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "x_n", "definition": "random(1..20)", "name": "c", "description": ""}, "N_2": {"templateType": "anything", "group": "Part b", "definition": "ceil((abs(b*c-a*d)-d*c*ep2)/(ep2*c^2))", "name": "N_2", "description": ""}, "b1": {"templateType": "anything", "group": "x_n", "definition": "random(1,-1)*random(2..9)", "name": "b1", "description": ""}, "a": {"templateType": "anything", "group": "x_n", "definition": "random(2..20)", "name": "a", "description": ""}, "ep2": {"templateType": "anything", "group": "Part b", "definition": "10^(-r2)", "name": "ep2", "description": ""}, "b": {"templateType": "anything", "group": "x_n", "definition": "if(b1*c=a*d,b1+1,b1)", "name": "b", "description": ""}, "r2": {"templateType": "anything", "group": "Part b", "definition": "r1+random(1,-1)", "name": "r2", "description": ""}, "tval": {"templateType": "anything", "group": "Part a", "definition": "(1 / c) * ((10 ^ r1 * abs(b * c -(a * d))) / c -d)", "name": "tval", "description": ""}, "N_1": {"templateType": "anything", "group": "Part a", "definition": "ceil((abs(b*c-a*d)-d*c*ep1)/(ep1*c^2))", "name": "N_1", "description": ""}}, "ungrouped_variables": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{N_1}", "minValue": "{N_1}", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "Find the least integer $N_1$ such that
\n\\[\\left|\\simplify[std]{x_n -({a} / {c})}\\right| \\le 10 ^ { -\\var{r1}}, \\textrm{ for } n \\geq N_1\\]
\nLeast $N_1=$ [[0]]
", "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{N_2}", "minValue": "{N_2}", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "Find the least integer $N_2$ such that
\n\\[\\left|\\simplify[std]{x_n -({a} / {c})}\\right| \\le 10 ^ {- \\var{r2}}, \\textrm{ for } n \\geq N_2\\]
\nLeast $N_2 = $ [[0]]
", "marks": 0}], "statement": "Let \\[x_n=\\simplify[std]{({a}n+{b})/({c}n+{d})}, \\quad n=1,\\; 2,\\; 3 \\ldots \\]
", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of a sequence", "limits", "MAS1601", "query", "sequences", "taking the limit", "tested1", "udf"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "4/07/2012:
\nChecked calculations.
\nSmall changes to Advice display.
\nLeft inequalities as $\\lt$.
\n21/07/2012:
\nAdded description.
\nAdded function chcop(a, b) to create coprime pairs - better display of solution.
\nThis finds an integer coprime to a in the range 1..20. b is set to a random value in the range.
\nChanged definition of variables a, b, c, d.
\n27/7/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n24/12/2012:
\nChanged $\\lt$ to $\\le$ and $\\gt$ t $\\ge$ throughout in order to be consistent with question 2 in this assignment. Added query tag to check on this.
\nChecked calculations, OK. Added tested1 tag.
\nAdded more information about the function chcop. Added udf tag.
\n", "licence": "Creative Commons Attribution 4.0 International", "description": "
$x_n=\\frac{an+b}{cn+d}$. Find the least integer $N$ such that $\\left|x_n -\\frac{a}{c}\\right| \\le 10 ^{-r},\\;n\\geq N$, $2\\leq r \\leq 6$.
"}, "variablesTest": {"condition": "gcd(a,c)=1 and gcd(c,d)=1", "maxRuns": 100}, "advice": "To find the least $N_1$ such that all terms from the the $N_1$th are within $10^{\\var{-r1}}$ of the limit we proceed as follows:
\n\\begin{align}
\\left| \\simplify[std]{x_n -({a} / {c})} \\right| \\le 10^{ -\\var{r1}} &\\iff \\left| \\simplify[std]{({a}n+{b})/({c}n+{d})-{a}/{c}}\\right| \\le 10 ^ { -\\var{r1}} \\\\
&\\iff \\simplify[std]{abs({b*c-a*d})/({c^2}n+{c*d})}\\le 10 ^ { -\\var{r1}}
\\end{align}
(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n+{c*d}} \\gt 0$, $\\forall n=1,\\; 2,\\; 3 \\ldots$)
\nRearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n+{c*d}}) \\times 10^{\\var{r1}}$ (this is positive and so the inequality does not reverse) we get:
\n\\[ \\simplify[std]{{c^2}n+{c*d}} \\ge \\var{10^r1*abs(b*c-a*d)} \\iff n \\ge \\frac{1}{\\var{c^2}}\\left(\\simplify[std]{{10^r1*abs(b*c-a*d)}-{c*d}}\\right) = \\var{tval} \\]
\nHence the least integer value is given by $N_1=\\var{N_1}$.
\nUsing the same method you should obtain $N_2=\\var{N_2}$.
"}], "name": "", "pickQuestions": 0}], "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions about the limits of sequences from a first year pure maths course.
"}, "type": "exam", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "browse": true, "showresultspage": "oncompletion", "preventleave": true, "allowregen": true, "showfrontpage": true}, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "pickQuestions": 0, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "extensions": [], "custom_part_types": [], "resources": []}