// Numbas version: finer_feedback_settings
{"name": "Sequences and Limits 1.1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["ab", "valexp", "v35", "al", "v15", "v150", "v610", "b1", "ga", "b3", "d2", "d1", "val", "s1", "v110", "v560", "s5", "v650", "k1", "r", "v45", "tol", "v210", "v250", "be", "v25", "v65", "de", "v450", "a3", "c1", "d", "a", "c", "b", "v550", "v350", "v510", "k", "v410", "n", "v55", "v310", "v660"], "name": "Sequences and Limits 1.1", "tags": ["examples of standard limits", "limit", "limits", "limits of sequences", "sequences", "standard limits", "taking the limit"], "preamble": {"css": "", "js": ""}, "advice": "<p>All calculations below are to $5$ decimal places.</p>\n<p>The notation $a \\approx b$ means that $a$ and $b$ are approximately equal.</p>\n<h4>a)</h4>\n<p>Using a calculator for $3$ values of $n$:</p>\n<table border=\"2\">\n  <thead>\n    <tr><th>$n$</th><th>$\\displaystyle{\\frac{1}{n^{1/\\var{r}}}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v15}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v110}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v150}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^{1/\\var{r}}}\\right)=0}$</p>\n<p>In fact $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^r}\\right)=0}$ for any $r \\gt 0$</p>\n<h4>b)</h4>\n<table border=\"2\">\n  <thead>\n    <tr><th>$n$</th><th>$\\displaystyle{\\var{k1}^{1/n}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v25}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v210}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v250}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\var{k1}^{1/n}=1}$, see next question as well.</p>\n<h4>c)</h4>\n<table border=\"2\">\n  <thead>   \n    <tr><th>$n$</th><th>$\\displaystyle{\\var{k}^{1/n}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v35}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v310}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v350}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\var{k}^{1/n}=1}$.</p>\n<p>From the last two questions it seems that $\\displaystyle{\\lim_{n \\to \\infty} k^{1/n}=1}$ for any $k \\gt 0$ &ndash; and this is fact true.</p>\n<h4>d)</h4>\n<table border=\"2\">\n<thead>\n  <tr><th>$n$</th><th>$\\displaystyle{\\frac{\\var{c}n+\\var{d}}{\\var{al}n-\\var{ga}}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$100$</td><td>$\\var{v45}$</td></tr>\n    <tr><td>$5000$</td><td>$\\var{v410}$</td></tr>\n    <tr><td>$5000000$</td><td>$\\var{v450}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that&nbsp; $\\displaystyle \\lim_{n \\to \\infty}\\left(\\simplify[std]{({c}n+{d})/({al}n-{ga})}\\right)\\;=\\;&nbsp;\\simplify[std]{{c}/{al}}$.</p>\n<p>In general <br/>\\[\\lim_{n \\to \\infty}\\left(\\frac{an+b}{cn+d}\\right)= \\frac{a}{c}\\] when $c \\neq 0$</p>\n<h4>e)</h4>\n<table border=\"2\">\n  <thead>\n    <tr><th>$n$</th><th>$\\displaystyle{\\left(\\simplify{{c}/{n}}\\right)^n}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$10$</td><td>$\\var{v55}$</td></tr>\n    <tr><td>$29$</td><td>$\\var{v510}$</td></tr>\n    <tr><td>$50$</td><td>$\\var{v550}$</td></tr>\n    <tr><td>$89$</td><td>$\\var{v560}$</td></tr>\n  </tbody>\n</table>\n<p>This indicates that $\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify{{c}/{n}}\\right)^n}= 0$. In general $\\displaystyle{\\lim_{n \\to \\infty} r^n= 0}$ if $|r| \\lt 1$</p>\n<h4>f)</h4>\n<p>We have the limit:<br/>\\[\\lim_{n\\to\\infty}\\left(1+\\frac{a}{n}\\right)=e^a\\]<br/>The following table confirms that the values are converging to (five decimal places) $\\displaystyle{\\simplify[std]{e^({a3}/{b3})={valexp}}}$</p>\n<table border=\"2\">\n<thead>\n  <tr><th>$n$</th><th>$\\displaystyle{\\left(\\simplify[std]{1+{a3}/({b3}n)}\\right)^n}$</th></tr>\n  </thead>\n  <tbody>\n    <tr><td>$10$</td><td>$\\var{v65}$</td></tr>\n    <tr><td>$100$</td><td>$\\var{v610}$</td></tr>\n    <tr><td>$1000$</td><td>$\\var{v650}$</td></tr>\n    <tr><td>$10000$</td><td>$\\var{v660}$</td></tr>\n  </tbody>\n</table>\n<p>Hence the answer asked for is $\\var{val}$ to $4$ decimal places.</p>\n<h4>g)</h4>\n<p>The answer to this question is based upon neglecting terms in polynomials in $n$ for large $n$.</p>\n<p>For example, $n^3+1000000n^2+1000000000 \\approx n^3$ for large $n$ as the $n^3$ term completely dominates the other terms as $n \\longrightarrow \\infty$.</p>\n<p>A more precise way of saying this is:<br/>\\[\\lim_{n\\to\\infty}\\left(\\frac{n^3+1000000n^2+1000000000}{n^3}\\right)=1\\]</p>\n<p>So for large $n$<br/>\\[\\begin{eqnarray*} \\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}}}&amp;\\approx&amp; \\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}}}\\\\ &amp;=&amp;\\frac{\\simplify[std]{{al^d}^(1/{d})n^{a}}} {\\simplify[std]{{ga^d1}^(1/{d1})n^{a}}}\\\\ &amp;=&amp;\\simplify[std]{{al}/{ga}} \\end{eqnarray*} \\]<br/>Hence the limit is $\\displaystyle{\\simplify[std]{{al}/{ga}}}$</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\frac{1}{n^{1/\\var{r}}}\\right)=\\;\\;}$[[0]]</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "0", "minValue": "0", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\var{k1}^{1/n}\\right)=\\;\\;}$[[0]]</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\var{k}^{1/n}\\right)=\\;\\;}$[[0]]</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n        <p>$\\displaystyle \\lim_{n \\to \\infty}\\left(\\simplify[std]{({c}n+{d})/({al}n-{ga})}\\right)\\;=\\;$[[0]]</p>\n      <p>Enter your answer as a fraction or integer, not as a decimal.</p>\n        ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Enter your answer as a fraction or integer, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{c}/{al}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify[std]{{c}/{n}}\\right)^n=\\;\\;}$[[0]]</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "0", "minValue": "0", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\left(\\simplify[std]{1+{a3}/({b3}n)}\\right)^n=\\;\\;}$[[0]]<br/>Input your answer to 4 decimal places.</p>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "val+tol", "minValue": "val-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>$\\displaystyle{\\lim_{n \\to \\infty}\\frac{\\left(\\simplify[std]{{al^d}n^({a*d})+{be}n^{b}+{c}}\\right)^{1/\\var{d}}}\n      \n      {\\left(\\simplify[std]{{ga^d1}n^({a*d1})+{de}n^{b1}+{c1}}\\right)^{1/\\var{d1}}}=\\;\\;}$[[0]]</p>\n      \n      \n      \n      <p>Enter your answer as a fraction or integer, not as a decimal.</p>\n      \n      \n        ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Enter your answer as a fraction or integer, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{al}/{ga}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>What are the following limits?</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ab": {"definition": "abs(a3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ab", "description": ""}, "valexp": {"definition": "precround(exp(a3/b3),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "valexp", "description": ""}, "v35": {"definition": "precround(k^(1/100),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v35", "description": ""}, "al": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "al", "description": ""}, "v15": {"definition": "precround(100^(-1/r),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v15", "description": ""}, "v150": {"definition": "precround(5000000^(-1/r),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v150", "description": ""}, "v610": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 100))) ^ 100,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v610", "description": ""}, "b1": {"definition": "a*d1-1", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "ga": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "ga", "description": ""}, "b3": {"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))", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "d2": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d1": {"definition": "if(d2=d,d+1,d2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "val": {"definition": "precround(exp(a3/b3),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "val", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "v110": {"definition": "precround(5000^(-1/r),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v110", "description": ""}, "v560": {"definition": "precround((c/n)^89,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v560", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "v650": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 1000))) ^ 1000,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v650", "description": ""}, "k1": {"definition": "random(100000..200000)", "templateType": "anything", "group": "Ungrouped variables", "name": "k1", "description": ""}, "v55": {"definition": "precround((c/n)^10,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v55", "description": ""}, "v45": {"definition": "precround((c * 100 + d) / (al * 100  -ga),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v45", "description": ""}, "tol": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "v210": {"definition": "precround(k1^(1/5000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v210", "description": ""}, "v250": {"definition": "precround(k1^(1/5000000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v250", "description": ""}, "be": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "be", "description": ""}, "v25": {"definition": "precround(k1^(1/100),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v25", "description": ""}, "v65": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 10))) ^ 10,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v65", "description": ""}, "de": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "de", "description": ""}, "v450": {"definition": "precround((c * 5000000 + d) / (al * 5000000  -ga),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v450", "description": ""}, "a3": {"definition": "s5*random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "c1": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "v350": {"definition": "precround(k^(1/5000000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v350", "description": ""}, "a": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s1*random(11..50)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "a*d-random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "v550": {"definition": "precround((c/n)^50,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v550", "description": ""}, "d": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "v510": {"definition": "precround((c/n)^29,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v510", "description": ""}, "k": {"definition": "random(2..20#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "k", "description": ""}, "v410": {"definition": "precround((c * 5000 + d) / (al * 5000  -ga),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v410", "description": ""}, "n": {"definition": "abs(c)+random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "r": {"definition": "random(2..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "v310": {"definition": "precround(k^(1/5000),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v310", "description": ""}, "v660": {"definition": "precround((1 + s5 * (abs(a3) / (b3 * 10000))) ^ 10000,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v660", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>4/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Improved display of prompt for fourth part.</p>\n        \t\t<p>Improved display of solution to fourth part.</p>\n        \t\t<p>Checked calculations.</p>\n        \t\t<p>No tolerance on answer to 6th part, got to be exact to 4dps. Tolerance variable, tol=0.</p>\n        \t\t<p><strong>21/07/2012:</strong></p>\n        \t\t<p>Added description.</p>\n        \t\t<p><strong>27/7/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Question appears to be working correctly.</p>\n        \t\t", "description": "<p>Seven standard elementary limits of sequences.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}