Question

Calcule o Limite :
a) $\lim_{x \to 1 ( \frac{ X^{n}-1 }{X-1} )$

Answer 1

Jr, para resolver o problema é necessário saber que $x^n-1=(x-1)(x^{n-1}+x^{n-2}+x^{n-3}+...+x^1+x^0)$.
 
 Isto posto,

$\lim_{x\rightarrow1}\left(\frac{x^n-1}{x-1}\right)=\\\\\\\lim_{x\rightarrow1}\left[\frac{(x-1)(x^{n-1}+x^{n-2}+x^{n-3}+...+x^1+1)}{x-1}\right]=\\\\\\\lim_{x\rightarrow1}(x^{n-1}+x^{n-2}+x^{n-3}+...+x^1+1)=\\\\\\1^{n-1}+1^{n-2}+1^{n-3}+...+1^1+1=\\\\\underbrace{1+1+1+...+1+1}_n=\\\\\boxed{n}$