Behaviour being used: Adaptive mode
Minimum fraction: 0
Question summary: The following statements are based on continuity of functions of several variables. Check the true statements : The function [f left parenthesis x comma y right parenthesis equals square root of open vertical bar xy close vertical bar end root] is not continuous at the origin; The function [f left parenthesis x comma y right parenthesis equals fraction numerator sin left parenthesis x squared y right parenthesis over denominator x squared plus y squared minus xy end fraction] cannot be continuously extended to the origin because the limit does not exist; The limit of the function [f colon real numbers squared minus open curly brackets open parentheses 0 comma 0 close parentheses close curly brackets rightwards arrow real numbers squared] defined by [f left parenthesis x comma y right parenthesis equals fraction numerator x open parentheses 1 minus cos left parenthesis y right parenthesis close parentheses over denominator x squared plus y squared end fraction] at the origin must be found employing the decomposition of the domain; The function [Error converting from MathML to accessible text.] does not reach absolute extrema on the set [Error converting from MathML to accessible text.]; The function [Error converting from MathML to accessible text.] is not continuous at the origin. It can be proved using directional limits; The limit of [Error converting from MathML to accessible text.] at the point [Error converting from MathML to accessible text.] equals -1/2 , hence the function is continuous
Right answer summary: The limit of the function [f colon real numbers squared minus open curly brackets open parentheses 0 comma 0 close parentheses close curly brackets rightwards arrow real numbers squared] defined by [f left parenthesis x comma y right parenthesis equals fraction numerator x open parentheses 1 minus cos left parenthesis y right parenthesis close parentheses over denominator x squared plus y squared end fraction] at the origin must be found employing the decomposition of the domain; The function [Error converting from MathML to accessible text.] is not continuous at the origin. It can be proved using directional limits
Question state: todo