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"], "name": "True/false statements about properties of continuity and limits, ", "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", "extensions": [], "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.
", "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/"}]}]}], "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/"}]}