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.

Hint: both numerator and denominator are divisible by $1-x$. — lulu, Mar 15 '17 at 21:34

score 3 · Accepted Answer · answered Mar 15 '17 at 21:35

3

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

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

answered Mar 15 '17 at 21:35

Simply Beautiful Art

74,685

How in the name of Jesus do you type in latex that fast? – mrnovice Mar 15 '17 at 21:36
1

@mrnovice Copy + paste – Simply Beautiful Art Mar 15 '17 at 21:36
Well played, +1 – mrnovice Mar 15 '17 at 21:36
I didn't come even close to thinking of geometric series. Thanks for the help. – Zerg Overmind Mar 15 '17 at 21:51
@ZergOvermind no problem! Glad to help :-) – Simply Beautiful Art Mar 15 '17 at 22:15

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

1 Answers1