# Nice limit

• Dec 24th 2009, 11:56 PM
dapore
Nice limit
• Dec 25th 2009, 12:12 AM
Prove It
Quote:

Originally Posted by dapore

Clearly the top and bottom both tend to 0 as $x \to 1$.

So you can use L'Hospital's Rule.

$\lim_{x \to 1}\frac{x^n - 1}{x^p - 1} = \lim_{x \to 1}\frac{\frac{d}{dx}(x^n - 1)}{\frac{d}{dx}(x^p - 1)}$

$= \lim_{x \to 1}\frac{nx^{n - 1}}{px^{p - 1}}$

$= \frac{n}{p}$.
• Dec 25th 2009, 12:39 AM
CaptainBlack
Quote:

Originally Posted by dapore

You should know that:

$x^k-1=(x-1)(x^{k-1}+x^{k-2}+ ... + x^2+x+1)$

CB
• Dec 30th 2009, 02:23 AM
dapore
Thank you my friends
• Dec 30th 2009, 02:57 AM
NonCommAlg
how would you solve the problem if $n,p$ were any non-zero real numbers? (of course L'Hospital's Rule is not allowed!) (Evilgrin)
• Dec 30th 2009, 07:17 PM
simplependulum
Quote:

Originally Posted by NonCommAlg
how would you solve the problem if $n,p$ were any non-zero real numbers? (of course L'Hospital's Rule is not allowed!) (Evilgrin)

This problem is equivalent to finding the derivative of $x^n$ from the first principle .

It is easy to solve when n is a rational number but how about irrational , or even transcendental ??
• Dec 31st 2009, 05:34 AM
Prove It
Quote:

Originally Posted by NonCommAlg
how would you solve the problem if $n,p$ were any non-zero real numbers? (of course L'Hospital's Rule is not allowed!) (Evilgrin)

Why can't you use L'Hospital? The numerator and denominator both tend to 0...
• Dec 31st 2009, 07:57 AM
simplependulum
Quote:

Originally Posted by Prove It
Why can't you use L'Hospital? The numerator and denominator both tend to 0...

It is a challenge he gives us , solving the limit without L'hospital , the difficuly rapidly increases !!
• Dec 31st 2009, 08:00 AM
Prove It
Couldn't you just perform long division?
• Dec 31st 2009, 08:03 AM
Moo
Quote:

Originally Posted by Prove It
Couldn't you just perform long division?

With n and p not being integers ?

But is it possible to use the 'definition' of the derivate number ?
• Dec 31st 2009, 08:06 AM
Prove It
Quote:

Originally Posted by Moo
With n and p not being integers ?

I don't see why not. The algebra could get quite messy though. I'm sure there's an easier way...
• Dec 31st 2009, 11:15 AM
NonCommAlg
here's a hint:
we only need to show that $\forall a \in \mathbb{R} : \ \lim_{x\to1} \frac{x^a - 1}{x-1} = a.$ it's easy to prove it for rational numbers. now for any $n \in \mathbb{N}$ choose $q_n \in \mathbb{Q}$ such that $q_n - \frac{1}{n} \leq a \leq q_n + \frac{1}{n}.$ then $\lim_{n\to\infty} q_n = a$ and ...
• Jan 1st 2010, 12:40 PM
Drexel28
Quote:

Originally Posted by NonCommAlg
here's a hint:
we only need to show that $\forall a \in \mathbb{R} : \ \lim_{x\to1} \frac{x^a - 1}{x-1} = a.$ it's easy to prove it for rational numbers. now for any $n \in \mathbb{N}$ choose $q_n \in \mathbb{Q}$ such that $q_n - \frac{1}{n} \leq a \leq q_n + \frac{1}{n}.$ then $\lim_{n\to\infty} q_n = a$ and ...

Yeah yeah, and then $\lim_{x\to1}\frac{x^a-1}{x^p-1}=\lim_{x\to1}\left\{\frac{x^a-1}{x-1}\cdot\frac{x-1}{x^p-1}\right\}=\frac{a}{p}$

By the way, was your point that if we can prove that the above limit is true for any $a\in\mathbb{Q}$ and since any real number has a sequence of rational points which converges to it we are done?