Límites de Función Seno

En esta lección vamos a ver algunos límites usando la función seno.

$stack l i m with x rightwards arrow 0 below space fraction numerator x minus s i n left parenthesis 3 x right parenthesis over denominator x plus s i n left parenthesis 2 x right parenthesis end fraction equals stack l i m with x rightwards arrow 0 below fraction numerator x open parentheses 1 minus fraction numerator 3 s i n left parenthesis 3 x right parenthesis over denominator 3 x end fraction close parentheses over denominator x open parentheses 1 plus fraction numerator 2 s i n left parenthesis 2 x right parenthesis over denominator 2 x end fraction close parentheses end fraction equals stack l i m with x rightwards arrow 0 below fraction numerator open parentheses 1 minus 3 left parenthesis 1 right parenthesis close parentheses over denominator open parentheses 1 plus 2 left parenthesis 1 right parenthesis close parentheses end fraction equals negative 2 over 3 stack l i m with x rightwards arrow 0 below space fraction numerator s i n left parenthesis 4 x right parenthesis over denominator s i n left parenthesis 2 x right parenthesis end fraction equals stack l i m with x rightwards arrow 0 below fraction numerator 4 x open parentheses fraction numerator s i n left parenthesis 4 x right parenthesis over denominator 4 x end fraction close parentheses over denominator 2 x open parentheses fraction numerator s i n left parenthesis 2 x right parenthesis over denominator 2 x end fraction close parentheses end fraction equals stack l i m with x rightwards arrow 0 below fraction numerator 4 open parentheses fraction numerator s i n left parenthesis 4 x right parenthesis over denominator 4 x end fraction close parentheses over denominator 2 open parentheses fraction numerator s i n left parenthesis 2 x right parenthesis over denominator 2 x end fraction close parentheses end fraction equals 4 over 2 equals 2 stack l i m with x rightwards arrow 0 below 1 over x sin left parenthesis x over 3 right parenthesis equals stack l i m with x rightwards arrow 0 below space fraction numerator sin left parenthesis x over 3 right parenthesis over denominator space 3 begin display style x over 3 end style end fraction equals 1 third stack l i m with x rightwards arrow 0 below space fraction numerator s i n left parenthesis x over 3 right parenthesis over denominator space x over 3 end fraction equals 1 third$

Nótese que unas de las propiedades que más se usó para resolver estos ejemplos fue:

$stack l i m with x rightwards arrow 0 below space fraction numerator sin left parenthesis f open square brackets x close square brackets right parenthesis over denominator f open square brackets x close square brackets end fraction equals 0 f open square brackets x close square brackets equals space x f open square brackets x close square brackets equals space x to the power of n space n greater than 1 f open square brackets x close square brackets equals space 1 over x to the power of n space n greater than 1$ $stack l i m with x rightwards arrow 0 below space fraction numerator sin left parenthesis f open square brackets x close square brackets right parenthesis over denominator f open square brackets x close square brackets end fraction equals 0 f open square brackets x close square brackets equals space x f open square brackets x close square brackets equals space x to the power of n space n greater than 1 f open square brackets x close square brackets equals space 1 over x to the power of n space n greater than 1$