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{"name": "Sequences: Limit Laws", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>The following questions deal with the limit laws of sequences.</p>\n<p></p>", "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["A", "B", "C", "D", "L", "f", "g", "list", "n0", "h", "j"], "parts": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "maxValue": "(A+f*B-C*D)/(L+g)", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "marks": 1, "correctAnswerFraction": true, "allowFractions": true, "minValue": "(A+f*B-C*D)/(L+g)", "mustBeReducedPC": 0}], "marks": 0, "prompt": "<p>Given that</p>\n<p>$\\displaystyle\\lim_{n\\rightarrow\\infty}a_n=\\var{A}$,&nbsp;<br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}b_n=\\var{B}$,&nbsp;<br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}c_n=\\var{C}$, <br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}d_n=\\var{D}$ and <br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}l_n=\\var{L}$.</p>\n<p>Determine the following limit:</p>\n<p style=\"text-align: center;\">$\\displaystyle\\lim_{n\\rightarrow\\infty} \\left(\\frac{a_n+\\var{f}b_n-c_nd_n}{l_n+\\var{g}}\\right)=$&nbsp;[[0]]</p>", "variableReplacements": [], "type": "gapfill"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "distractors": ["", "", "", ""], "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "matrix": [0, 0, 0, "1"], "minMarks": 0, "choices": ["<p>It diverges.</p>", "<p>It converges to some unknown number.</p>", "<p>There is insufficient&nbsp;information to tell if it converges or diverges.</p>", "<p>It converges to {h}.</p>"], "displayType": "dropdownlist", "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "marks": 0, "maxMarks": 0, "displayColumns": 0, "shuffleChoices": true}], "marks": 0, "prompt": "<p>Suppose $x_n\\le y_n \\le z_n$ for all $n\\ge \\var{n0}$, and $\\displaystyle\\lim_{n\\rightarrow\\infty}x_n=\\lim_{n\\rightarrow\\infty}z_n=\\var{h}$. Then what can be said about the sequence $\\{y_n\\}$?</p>\n<p>[[0]]</p>", "variableReplacements": [], "type": "gapfill"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "distractors": ["", "", "", "", ""], "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "matrix": [0, 0, 0, "1", 0], "minMarks": 0, "choices": ["<p>It diverges.</p>", "<p>It converges to some unknown number.</p>", "<p>There is insufficient&nbsp;information to tell if it converges or diverges.</p>", "<p>It converges to {j}.</p>", "<p>It converges to {j+0.5}.</p>"], "displayType": "dropdownlist", "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "marks": 0, "maxMarks": 0, "displayColumns": 0, "shuffleChoices": true}], "marks": 0, "prompt": "<p>Suppose $\\lim_{x\\rightarrow\\infty}f(x)=\\var{j}$ and $f(n)=a_n$ when $n$ is an integer. Then what can be said about the sequence $\\{a_n\\}$?</p>\n<p>[[0]]</p>", "variableReplacements": [], "type": "gapfill"}], "rulesets": {}, "variables": {"C": {"templateType": "anything", "name": "C", "group": "Ungrouped variables", "description": "", "definition": "list[2]"}, "D": {"templateType": "anything", "name": "D", "group": "Ungrouped variables", "description": "", "definition": "list[3]"}, "h": {"templateType": "anything", "name": "h", "group": "Ungrouped variables", "description": "", "definition": "list[7]"}, "j": {"templateType": "anything", "name": "j", "group": "Ungrouped variables", "description": "", "definition": "list[8]+0.5"}, "A": {"templateType": "anything", "name": "A", "group": "Ungrouped variables", "description": "", "definition": "list[0]"}, "L": {"templateType": "anything", "name": "L", "group": "Ungrouped variables", "description": "", "definition": "list[4]"}, "B": {"templateType": "anything", "name": "B", "group": "Ungrouped variables", "description": "", "definition": "list[1]"}, "g": {"templateType": "anything", "name": "g", "group": "Ungrouped variables", "description": "", "definition": "abs(list[6])"}, "f": {"templateType": "anything", "name": "f", "group": "Ungrouped variables", "description": "", "definition": "abs(list[5])"}, "list": {"templateType": "anything", "name": "list", "group": "Ungrouped variables", "description": "", "definition": "shuffle(-12..12 except 0)"}, "n0": {"templateType": "anything", "name": "n0", "group": "Ungrouped variables", "description": "", "definition": "random(10..1000#10)"}}, "name": "Sequences: Limit Laws", "variable_groups": [], "advice": "<p>a)&nbsp;</p>\n<p>The following steps through the application of the limit laws to our question</p>\n<p>$\\displaystyle\\begin{align}\\lim_{n\\rightarrow\\infty} \\left(\\frac{a_n+\\var{f}b_n-c_nd_n}{l_n+\\var{g}}\\right)&amp;=\\frac{\\lim_{n\\rightarrow\\infty}\\left(a_n+\\var{f}b_n-c_nd_n\\right)}{\\lim_{n\\rightarrow\\infty}\\left(l_n+\\var{g}\\right)}\\\\&amp;=\\frac{\\lim_{n\\rightarrow\\infty}(a_n)+\\lim_{n\\rightarrow\\infty}(\\var{f}b_n)-\\lim_{n\\rightarrow\\infty}(c_nd_n)}{\\lim_{n\\rightarrow\\infty}(l_n)+\\lim_{n\\rightarrow\\infty}(\\var{g})}\\\\&amp;=\\frac{\\lim_{n\\rightarrow\\infty}(a_n)+\\var{f}\\lim_{n\\rightarrow\\infty}(b_n)-\\lim_{n\\rightarrow\\infty}(c_n)\\lim_{n\\rightarrow\\infty}(d_n)}{\\lim_{n\\rightarrow\\infty}(l_n)+\\lim_{n\\rightarrow\\infty}(\\var{g})}\\\\&amp;=\\frac{\\var{A}+\\var{f}(\\var{B})-(\\var{C})(\\var{D})}{\\var{L}+\\var{g}}\\\\&amp;=\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{A+f*B-C*D}/{L+g}}\\end{align}$</p>\n<p></p>\n<p>b)&nbsp;</p>\n<p>The middle sequence,&nbsp;$\\{y_n\\}$, is squeezed by the sequence below, $\\{x_n\\}$, and the sequence above, $\\{z_n\\}$, which both converge to the same limit, therefore $\\{y_n\\}$ must also converge to the same limit. This is known as the squeeze theorem (for sequences).</p>\n<p></p>\n<p>c)&nbsp;</p>\n<p>The only difference between a function converging and a sequence converging is that the sequence is only defined for integers. So if&nbsp;the&nbsp;value of&nbsp;$a_n$ is the same as $f(n)$ and $f(x)$ converges to $\\var{j}$, then it stands to reason that the sequence $\\{a_n\\}$ would also converge to $\\var{j}$.</p>", "tags": [], "extensions": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Basic question on limit laws and the squeeze theorem for sequences.</p>"}, "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}