L´Hopital es el responsable…………!

Aplicamos L´Hopital en la resolución de un límite y nos hemos encontrado con una situación que no podemos resolver. Veamos que nos pasó:

Tomamos un número (x) y lo expresamos como diferencia de otros dos:

[math] x=b-a [/math]

Multiplicamos ambos miembros por ese mismo número, por lo tanto:

[math] x^2=\text{xb}-\text{xa} [/math]

Pero también se nos ocurrió elevar al cuadrado ambos miembros, entonces:

[math] x^2=(b-a)^2=b^2-2\text{ba}+a^2 [/math]

De modo que igualando lo igualable, nos resultó:

[math] \text{xb}-\text{xa}=b^2-2\text{ba}+a^2 [/math]

Digamos que podremos asociar términos con a, por un lado, y términos con b por el otro:

[math] \text{xb}-b^2+\text{ba}=a^2-\text{ba}+\text{xa} [/math]

Tal que:

[math] b\times (x-b+a)=a\times (a-b+x) [/math]

Y como tanto (x-b+a) como (a-b+x) resultan ser 0 a raíz de la hipótesis inicial hemos resuelto pasar al análisis matemático y escribir:

[math] b\times \frac{(x-b+a)}{(a-b+x)}=a\times \frac{(a-b+x)}{(x-b+a)} [/math]

De tal suerte que como ambas fracciones resultan ser [math] \frac{0}{0} [/math] se nos ocurrió aplicar límite con [math] x\rightarrow \infty [/math] por medio de L´Hopital dado que:

[math] \lim x [/math] [math] \rightarrow [/math] [math] \infty [/math] [math] \frac{f(x)}{g(x)}=\lim x\rightarrow \infty \frac{f(x)}{g(x)} [/math]

Y ambas derivadas dan 1 por lo tanto [math] b\times \frac{1}{1}=a\times \frac{1}{1} [/math] es decir:

[math] a=b [/math]

Por lo tanto "un par de números cualesquiera siempre son iguales"

Su voto: Nada


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