Question

Calcule o limite sem usar l hopital

Limite x tendendo a 1.
X^950 - 1 / x -1

Answer 1

Explicação passo-a-passo:

$\sf lim_{x~\Rightarrow~1}~\dfrac{x^{950}-1}{x-1}$

Veja que:

• $\sf x^2-1=(x-1)\cdot(x+1)$

• $\sf x^4-1=(x-1)\cdot(x^3+x^2+x+1)$

• $\sf x^6-1=(x-1)\cdot(x^5+x^4+x^3+x^2+x+1)$

Assim:

$\sf x^{950}-1=(x-1)\cdot(x^{949}+x^{948}+\dots+x+1)$

Temos:

$\sf lim_{x~\Rightarrow~1}~\dfrac{x^{950}-1}{x-1}$

$\sf =lim_{x\Rightarrow~1}~\dfrac{(x-1)\cdot(x^{949}+x^{948}+\dots+x+1)}{x-1}$

$\sf =lim_{x~\Rightarrow~1}~(x^{949}+x^{948}+\dots+x+1)$

$\sf =1+1+1+\dots+1+1$

$\sf =950\cdot1$

$\sf =\red{950}$