Question

Ache o limite sem utilizar L'Hospital:
So consigo utilizando L'Hospital, heeelp$\lim_{x \to \1} \(\frac{\sqrt[3]{x^{2}}-2\sqrt[3]{x}+1 }{(x-1)^{2} }$

Answer 1

Resposta:

1/9

Explicação passo-a-passo:

Temos que:

lim {raiz3(x^2) -2.raiz3(x) +1} / (x-1)^2

x->1

lim {(x^2)^(1/3) -2.(x^(1/3)) +1} / (x-1)^2

x->1

lim {x^(2/3) -2.x^(1/3) +1} / (x-1)^2

x->1

lim {[x^(1/3)]^2 -2.[x^(1/3)] +1} / (x-1)^2

x->1

Fazendo x^(1/3) = w, temos que:

• [x^(1/3)]^3 = w^3 => x= w^3
• Se x->1 => w->1

Logo:

lim {w^2 -2.w +1} / (w^3 -1)^2

w->1

lim (w - 1)^2 / (w^3 -1)^2

w->1

lim {(w - 1)/(w^3 -1)}^2

w->1

Pelo Método de Briot-Ruffini temos que:

w^3 +0.w^2 +0.w -1 | w - 1

-w^3 + w^2 ——————

-------------------- w^2 +w + 1

0 w^2 + 0.w

-w^2 + w

----------------

0 w - 1

-w + 1

----------

0 0

Logo:

lim {(w - 1)/(w^3 -1)}^2

w->1

lim {1/(w^2 + w + 1)}^2

w->1

lim 1/[(w^2 + w + 1)^2]

w->1

Assim, temos:

1/[(1^2 + 1 + 1)^2]

1/[(3)^2]

1/9

Blz?

Abs :)