How do I calculate this limit $\lim\limits_{x \to 0}\frac{ 1-x^n}{1-x^m}$?

Question

I'm having problem finding limit of lim $x \to 1$ for $(1-x^n)/(1-x^m$). I know that the result is $n/m$ but I can't really come up with how to modify the expression to arrive at that.

Answer 1

If $n,m\in\mathbb N$, then

$$1-x^n=(1-x)(1+x+x^2+\dots+x^{n-1})$$

Thus, your ratio reduces to

$$\frac{1-x^n}{1-x^m}=\frac{1+x+x^2+\dots+x^{n-1}}{1+x+x^2+\dots+x^{m-1}}\stackrel{x\to1}\longrightarrow\frac nm$$