Limits of m/n as x Approaches 1

[spartas]

lim x^m-1/xⁿ-1 m,n elements of N
x→1
the answer is m/n but i have no idea how to start or solve this!

 

[MarkFL]

Consider the following:

\(\displaystyle \sum_{k=0}^n\left(x^k\right)=\frac{x^{n+1}-1}{x-1}\)

Can you now rewrite the limit to get a determinate form?

 

[MarkFL]

We are given to find:

\(\displaystyle L=\lim_{x\to1}\frac{x^m-1}{x^n-1}\) where \(\displaystyle m,n\in\mathbb{N}\)

Using the hint I suggested, we may write:

\(\displaystyle x^m-1=(x-1)\sum_{k=0}^{m-1}\left(x^k\right)\)

\(\displaystyle x^n-1=(x-1)\sum_{k=0}^{n-1}\left(x^k\right)\)

And so our limit becomes:

\(\displaystyle L=\lim_{x\to1}\frac{(x-1)\sum\limits_{k=0}^{m-1}\left(x^k\right)}{(x-1)\sum\limits_{k=0}^{n-1}\left(x^k\right)}=\lim_{x\to1}\frac{\sum\limits_{k=0}^{m-1}\left(x^k\right)}{\sum\limits_{k=0}^{n-1}\left(x^k\right)}=\frac{\sum\limits_{k=0}^{m-1}\left(1\right)}{\sum\limits_{k=0}^{n-1}\left(1\right)}=\frac{\sum\limits_{k=1}^{m}\left(1\right)}{\sum\limits_{k=1}^{n}\left(1\right)}=\frac{m}{n}\)