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:
\\[\\begin{eqnarray*} \\left|\\simplify[std]{x_n -({a} / {c})}\\right| \\leq 10 ^ { -\\var{r}} &\\Leftrightarrow&\\left|\\simplify[std]{({a}n+{b})/({c}n+{d})-{a}/{c}}\\right| \\leq 10 ^ { -\\var{r}}\\\\ &\\Leftrightarrow&\\simplify[std]{abs({b*c-a*d})/({c^2}n+{c*d})}\\leq 10 ^ { -\\var{r}} \\end{eqnarray*} \\]
(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n+{c*d}} \\gt 0,\\;\\;\\forall n=1,\\;2,\\;3\\ldots$)
Rearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n+{c*d}})10^{\\var{r}}$ (this is positive and so the inequality does not reverse) we get:
\\[\\simplify[std]{{c^2}n+{c*d}} \\geq \\var{10^r*abs(b*c-a*d)} \\Leftrightarrow n \\geq \\simplify[std]{{1}/{c^2}({10^r*abs(b*c-a*d)}-{c*d})}=\\var{tval}\\]
Hence the least integer value is given by rounding up i.e. $N=\\var{N}$.
", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"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 ", "marks": 0, "gaps": [{"notallowed": {"message": "Enter your answer as a fraction or integer, not as a decimal.
", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a}/{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "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 ", "marks": 0, "gaps": [{"displayColumns": 0, "matrix": [1, 0, 0, 0, 0], "shuffleChoices": true, "maxMarks": 0, "distractors": ["", "", "", "", ""], "choices": ["{N}
", "{N1}
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", "{N3}
", "{N4}
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", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "chcop(a,a)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "if(a*d=b1*c,b1+1,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "chcop(c,c)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "s": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "r": {"definition": "random(2,3,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "b1": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "n": {"definition": "ceil((abs(b*c-a*d)-d*c*ep)/(ep*c^2))", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "t": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "n1": {"definition": "if(N>100,N+s2*random(8..52#4),if(N>50,N+s2*random(8..48#4),N+random(1,3)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "if(N>100,N+s3*random(7..43#4),if(N>50,N+s3*random(7..43#4),N+random(2,4)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "n3": {"definition": "if(N>100,N+s4*random(6..42#4),if(N>50,N+s4*random(6..42#4),N+random(5,6)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n3", "description": ""}, "n4": {"definition": "if(N>100,N+s5*random(5..41#4),if(N>50,N+s5*random(5..41#4),N+random(7,8)))", "templateType": "anything", "group": "Ungrouped variables", "name": "n4", "description": ""}, "ep": {"definition": "10^(-r)", "templateType": "anything", "group": "Ungrouped variables", "name": "ep", "description": ""}, "tval": {"definition": "(1 / c) * ((10 ^ r * abs(b * c -(a * d))) / c -d)", "templateType": "anything", "group": "Ungrouped variables", "name": "tval", "description": ""}}, "metadata": {"notes": "\n \t\t4/07/2012:
\n \t\t
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:
\n \t\tAdded description.
\n \t\tNeeds better tags to describe second part.
\n \t\tAlso 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 \t\t(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.
\n \t\tGot rid of the changes needed tag.
\n \t\t27/7/2012:
\n \t\tAdded tags.
\n \t\t", "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 $|x_n-L| \\lt 10^{-r},\\;n \\geq N,\\;r \\in \\{2,\\;3,\\;4\\}$.
", "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/"}]}