Evaluate $\lim\limits_{x\to\ 1}\frac{x+x^2+\cdots+x^n-n}{x-1}$ without L'Hospital's rule

Question

I used substitution $$t=x-1, x=t+1,x\rightarrow1\Rightarrow t\rightarrow 0$$

Now the expression is $$\lim_{t\to\ 0}\frac{t+1+(t+1)^2+\cdots+(t+1)^n-n}{t}$$

Can we use the sum of geometric sequence or squeeze theorem?

Answer 1

Hint: the numerator can be written:

$$(x-1)+(x^2-1) +\cdots + (x^n-1)$$