Método de Evaluación de Límites. Indeterminaciones IV

A parte de las indeterminaciones estudiadas, existen otras de diversa índole que aparecen en la resolución de ciertos límites que sobrepasan la forma tradicional de resolución. Veremos varios de esos límites esta página.

Ejemplos.

$f left parenthesis x right parenthesis equals fraction numerator x plus 3 over denominator x squared minus 25 end fraction space minus space fraction numerator x over denominator x minus 5 end fraction stack l i m with x rightwards arrow 5 below space f left parenthesis x right parenthesis equals stack l i m with x rightwards arrow 5 below space fraction numerator x plus 3 over denominator x squared minus 25 end fraction space minus space fraction numerator x over denominator x minus 5 end fraction equals stack l i m with x rightwards arrow 5 below space fraction numerator 5 plus 3 over denominator 5 squared minus 25 end fraction space minus space fraction numerator 5 over denominator 5 minus 5 end fraction equals stack l i m with x rightwards arrow 5 below space 8 over 0 space minus space 5 over 0 stack l i m with x rightwards arrow 5 below space 8 over 0 space minus space 5 over 0 equals 8 over 0 space minus space 5 over 0 equals infinity minus infinity I n d e t e r m i n a c i o n space d e l space t i p o space infinity minus infinity. g left parenthesis x right parenthesis equals fraction numerator square root of x squared minus 2 end root space minus space square root of x squared plus x end root space over denominator x space plus 3 end fraction equals fraction numerator square root of x squared minus 2 end root space over denominator x space plus 3 end fraction minus space fraction numerator square root of x squared plus x end root space over denominator x space plus 3 end fraction stack l i m with x rightwards arrow negative 3 below g left parenthesis x right parenthesis equals stack l i m with x rightwards arrow negative 3 below fraction numerator square root of x squared minus 2 end root space minus space square root of x squared plus x end root space over denominator x space plus 3 end fraction equals stack l i m with x rightwards arrow negative 3 below fraction numerator square root of x squared minus 2 end root space over denominator x space plus 3 end fraction minus space fraction numerator square root of x squared plus x end root space over denominator x space plus 3 end fraction equals stack l i m with x rightwards arrow negative 3 below fraction numerator square root of left parenthesis negative 3 right parenthesis squared minus 2 end root space over denominator left parenthesis negative 3 right parenthesis space plus 3 end fraction minus space fraction numerator square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis negative 3 right parenthesis end root space over denominator left parenthesis negative 3 right parenthesis space plus 3 end fraction stack l i m with x rightwards arrow negative 3 below fraction numerator square root of left parenthesis negative 3 right parenthesis squared minus 2 end root space over denominator left parenthesis negative 3 right parenthesis space plus 3 end fraction minus space fraction numerator square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis negative 3 right parenthesis end root space over denominator left parenthesis negative 3 right parenthesis space plus 3 end fraction equals fraction numerator square root of left parenthesis negative 3 right parenthesis squared minus 2 end root space over denominator 0 end fraction minus space fraction numerator square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis negative 3 right parenthesis end root space over denominator 0 end fraction equals infinity minus infinity. I n d e t e r m i n a c i o n space d e l space t i p o space infinity minus infinity. h left parenthesis x right parenthesis equals square root of x squared plus 3 x end root space minus space x stack l i m with x rightwards arrow infinity below h left parenthesis x right parenthesis equals stack l i m with x rightwards arrow infinity below space space square root of x squared plus 3 x end root space minus space x equals space stack l i m with x rightwards arrow infinity below space space square root of left parenthesis infinity right parenthesis squared plus 3 left parenthesis infinity right parenthesis end root space minus space left parenthesis infinity right parenthesis equals infinity minus infinity. I n d e t e r m i n a c i o n space d e l space t i p o space infinity minus infinity.$ $j left parenthesis x right parenthesis equals open square brackets x squared plus x plus 1 close square brackets to the power of 1 over x end exponent stack l i m with x rightwards arrow 0 below space j left parenthesis x right parenthesis equals stack l i m with x rightwards arrow 0 below space open square brackets x squared plus x plus 1 close square brackets to the power of 1 over x end exponent equals stack l i m with x rightwards arrow 0 below open square brackets 0 squared plus 0 plus 1 close square brackets to the power of 1 over 0 end exponent equals 1 to the power of infinity I n d e t e r m i n a c i o n space d e l space t i p o space 1 to the power of infinity$

$w left parenthesis x right parenthesis equals open parentheses fraction numerator x squared minus x plus 1 over denominator x squared plus 1 end fraction close parentheses to the power of fraction numerator x squared plus 1 over denominator x end fraction end exponent stack l i m with x rightwards arrow 0 below space w left parenthesis x right parenthesis equals stack l i m with x rightwards arrow 0 below space open parentheses fraction numerator x squared minus x plus 1 over denominator x squared plus 1 end fraction close parentheses to the power of fraction numerator x squared plus 1 over denominator x end fraction end exponent equals stack l i m with x rightwards arrow 0 below space open parentheses fraction numerator 0 squared minus 0 plus 1 over denominator 0 squared plus 1 end fraction close parentheses to the power of fraction numerator 0 squared plus 1 over denominator 0 end fraction end exponent equals 1 to the power of infinity i n d e t e r m i n a c i o n space d e l space t i p o space 1 to the power of infinity$

Nótese que en todos los ejemplos ilustrados la variable x tiende a un número finito o a cero. Esta es una parte de todas las formas de límite que existen. En realidad también la variable x puede tender a $infinity$ . Estos ejemplos los veremos en la próxima lección.