Método de Evaluación. Indeterminaciones III

En la lección anterior se abordaron dos tipo de indeterminación. Repitiendo una de las ideas fundamentales de la pasada clase, es aconsejable en esos casos, primeramente utilizar el método de evaluación tradicional para determinar a qué tipo de indeterminación nos estamos enfrentando. En esta lección abordaremos las del tipo $0 to the power of 0$ y $infinity to the power of 0$ .

Ejemplos.

Halle los siguientes límites de funciones o intente determinar frente a qué tipo de indeterminación se encuentra:

$f left parenthesis x right parenthesis equals x to the power of sin open parentheses x close parentheses end exponent stack l i m with x rightwards arrow 0 below space f left parenthesis x right parenthesis equals stack l i m with x rightwards arrow 0 below space x to the power of s i n open parentheses x close parentheses end exponent equals stack l i m with x rightwards arrow 0 below space 0 to the power of s i n open parentheses 0 close parentheses end exponent equals 0 to the power of 0 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 0 to the power of 0 g left parenthesis x right parenthesis equals tan open parentheses x close parentheses to the power of left parenthesis x squared plus space 3 x right parenthesis end exponent stack l i m with x rightwards arrow 0 below space g left parenthesis x right parenthesis equals stack l i m with x rightwards arrow 0 below space t a n open parentheses x close parentheses to the power of left parenthesis x squared plus space 3 x right parenthesis end exponent equals stack l i m with x rightwards arrow 0 below space t a n open parentheses x close parentheses to the power of left parenthesis x squared plus space 3 x right parenthesis end exponent equals 0 to the power of 0 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 0 to the power of 0 h left parenthesis x right parenthesis equals left parenthesis e to the power of x squared end exponent space minus space 1 right parenthesis to the power of 2 x end exponent stack l i m with x rightwards arrow 0 below space h left parenthesis x right parenthesis equals stack l i m with x rightwards arrow 0 below space left parenthesis e to the power of x squared end exponent space minus space 1 right parenthesis to the power of 2 x end exponent equals stack l i m with x rightwards arrow 0 below space left parenthesis e to the power of left parenthesis 0 right parenthesis squared end exponent space minus space 1 right parenthesis to the power of 2 left parenthesis 0 right parenthesis end exponent equals left parenthesis e to the power of left parenthesis 0 right parenthesis end exponent space minus space 1 right parenthesis to the power of 0 equals left parenthesis 1 space minus space 1 right parenthesis to the power of 0 equals 0 to the power of 0 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 0 to the power of 0 j left parenthesis x right parenthesis equals 1 over x to the power of sin open parentheses x close parentheses 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 1 over x to the power of s i n open parentheses x close parentheses end exponent equals stack l i m with x rightwards arrow 0 below space 1 over 0 to the power of s i n open parentheses 0 close parentheses end exponent equals infinity to the power of 0 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 to the power of 0 k left parenthesis x right parenthesis equals fraction numerator 1 over denominator e to the power of x minus 1 end fraction to the power of left parenthesis x cubed plus space 3 e to the power of x space minus 3 right parenthesis 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 fraction numerator 1 over denominator e to the power of x minus 1 end fraction to the power of left parenthesis x cubed plus space 3 e to the power of x space minus 3 right parenthesis end exponent equals stack l i m with x rightwards arrow 0 below space fraction numerator 1 over denominator e to the power of 0 minus 1 end fraction to the power of left parenthesis 0 cubed plus space 3 e to the power of 0 space minus 3 right parenthesis end exponent equals stack l i m with x rightwards arrow 0 below space fraction numerator 1 over denominator e to the power of 0 minus 1 end fraction to the power of left parenthesis space 3 e to the power of 0 space minus 3 right parenthesis end exponent equals stack l i m with x rightwards arrow 0 below space fraction numerator 1 over denominator 1 minus 1 end fraction to the power of left parenthesis space 3 space minus 3 right parenthesis end exponent equals stack l i m with x rightwards arrow 0 below space 1 over 0 to the power of left parenthesis space 0 right parenthesis end exponent equals infinity to the power of 0 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 to the power of 0 s left parenthesis x right parenthesis equals 1 over x to the power of x stack l i m with x rightwards arrow 0 below space s left parenthesis x right parenthesis equals stack l i m with x rightwards arrow 0 below space 1 over x to the power of x equals stack l i m with x rightwards arrow 0 below space 1 over 0 to the power of 0 equals infinity to the power of 0 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 to the power of 0$