Método de Factorización

Factorizar funciones a las cuales queremos determinar el limite puede ser un procedimiento muy útil; consiste en aplicar los productos notables, las propiedades de la multiplicación, división y demás operaciones básicas y no tan básicas a la resolución de limites.

Ejemplos:

$f left parenthesis x right parenthesis equals fraction numerator 2 minus square root of x over denominator x minus 4 end fraction stack l i m with x rightwards arrow 4 below space fraction numerator 2 minus square root of x over denominator x minus 4 end fraction equals stack l i m with x rightwards arrow 4 below space fraction numerator negative open parentheses square root of x minus 2 close parentheses over denominator x minus 4 end fraction equals stack l i m with x rightwards arrow 4 below space fraction numerator negative open parentheses square root of x minus 2 close parentheses over denominator x minus 4 end fraction equals stack l i m with x rightwards arrow 4 below space fraction numerator negative open parentheses square root of x minus 2 close parentheses over denominator open parentheses square root of x minus 2 close parentheses open parentheses square root of x plus 2 close parentheses end fraction stack l i m with x rightwards arrow 4 below space fraction numerator negative open parentheses square root of x minus 2 close parentheses over denominator open parentheses square root of x minus 2 close parentheses open parentheses square root of x plus 2 close parentheses end fraction equals stack l i m with x rightwards arrow 4 below space fraction numerator negative 1 over denominator 1 open parentheses square root of x plus 2 close parentheses end fraction equals stack l i m with x rightwards arrow 4 below space fraction numerator negative 1 over denominator 1 open parentheses square root of 4 plus 2 close parentheses end fraction equals stack l i m with x rightwards arrow 4 below space fraction numerator negative 1 over denominator 1 open parentheses 2 plus 2 close parentheses end fraction equals stack l i m with x rightwards arrow 4 below space fraction numerator negative 1 over denominator 4 end fraction equals fraction numerator negative 1 over denominator 4 end fraction x minus 4 equals open parentheses square root of open parentheses x close parentheses end root close parentheses squared minus 2 squared equals open parentheses square root of x minus 2 close parentheses open parentheses square root of x plus 2 close parentheses g left parenthesis x right parenthesis equals fraction numerator 1 minus 9 x squared over denominator 3 x minus 1 end fraction stack l i m with x rightwards arrow 1 third below space fraction numerator 1 minus 9 x squared over denominator 3 x minus 1 end fraction equals stack l i m with x rightwards arrow 1 third below space fraction numerator 1 minus open parentheses 3 x close parentheses squared over denominator 3 x minus 1 end fraction equals stack l i m with x rightwards arrow 1 third below space fraction numerator open parentheses 1 minus 3 x close parentheses open parentheses 1 plus 3 x close parentheses over denominator 3 x minus 1 end fraction equals stack l i m with x rightwards arrow 1 third below space fraction numerator negative open parentheses 3 x minus 1 close parentheses open parentheses 1 plus 3 x close parentheses over denominator 3 x minus 1 end fraction equals stack l i m with x rightwards arrow 1 third below space fraction numerator negative 1 open parentheses 1 plus 3 x close parentheses over denominator 1 end fraction stack l i m with x rightwards arrow 1 third below space fraction numerator negative 1 open parentheses 1 plus 3 x close parentheses over denominator 1 end fraction equals stack l i m with x rightwards arrow 1 third below space open square brackets negative open parentheses 1 plus 3 x close parentheses close square brackets space equals stack l i m with x rightwards arrow 1 third below space open square brackets negative open parentheses 1 plus 3 open parentheses 1 third close parentheses close parentheses close square brackets equals stack l i m with x rightwards arrow 1 third below space open square brackets negative open parentheses 1 plus 1 close parentheses close square brackets equals negative 2 h left parenthesis x right parenthesis equals fraction numerator x cubed minus 1 over denominator x minus 1 end fraction equals fraction numerator open parentheses x minus 1 close parentheses open parentheses x squared plus x plus 1 close parentheses over denominator x minus 1 end fraction equals open parentheses x squared plus x plus 1 close parentheses stack l i m with x rightwards arrow 1 below space fraction numerator x cubed minus 1 over denominator x minus 1 end fraction equals stack l i m with x rightwards arrow 1 below space space fraction numerator open parentheses x minus 1 close parentheses open parentheses x squared plus x plus 1 close parentheses over denominator x minus 1 end fraction equals stack l i m with x rightwards arrow 1 below space open parentheses x squared plus x plus 1 close parentheses equals 3 j left parenthesis x right parenthesis equals fraction numerator x cubed plus 8 over denominator x plus 2 end fraction equals fraction numerator open parentheses x plus 2 close parentheses open parentheses x squared minus 2 x plus 4 close parentheses over denominator x plus 2 end fraction equals open parentheses x squared minus 2 x plus 4 close parentheses stack l i m with x rightwards arrow negative 2 below fraction numerator x cubed plus 8 over denominator x plus 2 end fraction equals stack l i m with x rightwards arrow negative 2 below space fraction numerator open parentheses x plus 2 close parentheses open parentheses x squared minus 2 x plus 4 close parentheses over denominator x plus 2 end fraction equals stack l i m with x rightwards arrow negative 2 below space open parentheses x squared minus 2 x plus 4 close parentheses equals 4 plus 4 plus 4 equals 12$