Question

Limite de (raiz cúbica de x)-1/(raiz quarta de x)-1 quando x tende a 1

Answer 1

Resposta:

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(a-b)³=a³-3a²b+3ab²-b³

(a-b)³=a³-b³-3ab*(a-b)

a³-b³=(a-b)*[(a-b)²+3ab]

a³-b³=(a-b)*(a²+b²+ab)

(a-b)=(a³-b³)/(a²+b²+ab)

Se a=∛x  e b=1

∛x -1 =(x-1)/(∛x²+1+∛x)

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(∜x-1)*(∜x+1)=√x-1   ==>(∜x-1) =(√x-1)/(∜x+1)

(√x-1)*(√x+1) =x-1   ==>(√x-1)=(x-1)/(√x+1)

então:

(∜x-1) = (x-1)/[(∜x+1)*(√x+1)]

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Lim  (∛x-1)/(∜x -1)

x-->1

Lim [ (x-1)/(∛x²+1+∛x)] / [(x-1)/[(∜x+1)*(√x+1)]]

x-->1

Lim [ (x-1)/(∛x²+1+∛x)] * [[(∜x+1)*(√x+1)]/(x-1)]

x-->1

Lim [ 1/(∛x²+1+∛x)] * [[(∜x+1)*(√x+1)]/1]

x-->1

= [ 1/(∛1²+1+∛1)] * [[(∜1+1)*(√1+1)]/1]

= [ 1/(1+1+1)] * [[(1+1)*(1+1)]/1]

=(1/3) * (2*2)/1  =4/3