Question

Lim
X-1

x3-4x² +5x-2/x³-x²-x+1​

Answer 1

                                            Limites

Respuesta:   $-\frac{1}{2}$

Explicación paso a paso:

$\lim _{x\to \:1}\left(\frac{x^3-4x^2+5x-2}{x^3-x^2-x+1}\right)$

$\mathrm{Simplificar}\:$

$\mathrm{Factorizar}\:x^3-4x^2+5x-2:\quad$

$x^3-4x^2+5x-2$

$\mathrm{Utilizar\:el\:teorema\:de\:la\:ra\'iz\:racional}$

$a_0=2,\:\quad a_n=1$

$\mathrm{Los\:divisores\:de\:}a_0:\quad 1,\:2,\:\quad \mathrm{Los\:divisores\:de\:}a_n:\quad 1$

$\mathrm{Por\:lo\:tanto,\:verificar\:los\:siguientes\:numeros\:racionales:\quad }\pm \frac{1,\:2}{1}$

$\frac{1}{1}\mathrm{\:es\:la\:ra\'iz\:de\:la\:expresi\'on,\:por\:lo\:tanto,\:factorizar\:}x-1$

$=\left(x-1\right)\frac{x^3-4x^2+5x-2}{x-1}$

$\frac{x^3-4x^2+5x-2}{x-1}=x^2-3x+2$

$\mathrm{Factorizar}\:x^2-3x+2:\quad$

$=\left(x-1\right)\left(x-1\right)\left(x-2\right)$

$=\left(x-1\right)^2\left(x-2\right)$

Queda:

$=\frac{\left(x-1\right)^2\left(x-2\right)}{x^3-x^2-x+1}$

$\mathrm{Factorizar}\:x^3-x^2-x+1:\quad$

$=\left(x^3-x^2\right)+\left(-x+1\right)$

$\mathrm{Factorizar\:}-1\mathrm{\:de\:}-x+1\mathrm{:\quad }-\left(x-1\right)$

$\mathrm{Factorizar\:}x^2\mathrm{\:de\:}x^3-x^2\mathrm{:\quad }x^2\left(x-1\right)$

$=-\left(x-1\right)+x^2\left(x-1\right)$

$\mathrm{Factorizar\:el\:termino\:com\'un\:}x-1$

$=\left(x-1\right)\left(x^2-1\right)$

$\mathrm{Factorizar}\:x^2-1:\quad \left(x+1\right)\left(x-1\right)$

$=\left(x-1\right)\left(x+1\right)\left(x-1\right)$

$=\left(x-1\right)^2\left(x+1\right)$

$\mathrm{Queda:}$

$=\frac{\left(x-1\right)^2\left(x-2\right)}{\left(x-1\right)^2\left(x+1\right)}$

$\mathrm{Eliminar\:los\:terminos\:comunes:}\:\left(x-1\right)^2$

$=\frac{x-2}{x+1}$

$\mathrm{Entonces}$

$=\lim _{x\to \:1}\left(\frac{x-2}{x+1}\right)$

$\mathrm{Sustituir\:la\:variable}$

$=\frac{1-2}{1+1}$

$\mathrm{Simplificar}$

$=-\frac{1}{2}$       ✔️ Respuesta

Saludos