Prove that for all $n \gt 1$ $\lim_{r \to 1} \frac {r^n-1}{r-1} = n$

Question

Prove that for all numbers $n \gt 1$ $\lim_{r \to 1} \frac {r^n-1}{r-1} = n$

I think induction will work for this, but I can't seem to figure it out. I have the base $n=1$ case because $\lim_{r \to 1} \frac {r^1-1}{r-1} = 1$ holds, then by induction hypothesis I would say it hold for $n$, then I just need to show it hold for $(n+1)$, which is where I'm stuck.

Answer 1

Hint: $(r^n-1)=(r^{n-1}+\cdots+r+1)(r-1)$.

Answer 2

Let $f(r) =r^n $. $f'(r) =nr^{n-1} $.

But $f'(r) =\lim_{x \to r} \frac{f(x)-f(r)}{x-r} =\lim_{x \to r} \frac{x^n-r^n}{x-r} $ so $n =f'(1) =\lim_{x \to 1} \frac{x^n-1}{x-1} $