Without L.R.,Long division,definition of derivative and eplison-delta prove, evaluate

**General** · January 13th 2010, 12:19 PM

Without what I said, Evaluate :
$\lim_{x \rightarrow 1} \frac{x^5-1}{x-1}$

**Jhevon** · January 13th 2010, 12:26 PM

Originally Posted by General

Without what I said, Evaluate :
$\lim_{x \rightarrow 1} \frac{x^5-1}{x-1}$

factor: $x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1)$ , the bad guys cancel, evaluate at 1 and you're done.

**General** · January 13th 2010, 12:27 PM

Originally Posted by Jhevon

factor: $x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1)$ , the bad guys cancel, evaluate at 1 and you're done.

How did you factor it ?
1 is zero of $x^5-1$
then (x-1) is a factor .. and you find $x^4+x^3+x^2+x+1$ by long division
which is not allowed to use.

**Jhevon** · January 13th 2010, 12:32 PM

Originally Posted by General

How do you factor it ?

It's a known fact that $a^n - b^n = (a - b)(a^{n - 1} + a^{n - 2}b + a^{n - 3}b^2 + \cdots + ab^{n - 2} + b^{n - 1})$ when $n$ is an odd natural number

**General** · January 13th 2010, 12:33 PM

Originally Posted by Jhevon

It's a known fact that $a^n - b^n = (a - b)(a^{n - 1} + a^{n - 2}b + a^{n - 3}b^2 + \cdots + ab^{n - 2} + b^{n - 1})$ when $n$ is an odd natural number

OK.
Waiting for the other solutions.
I want to see another ideas

Thanks.

**Jhevon** · January 13th 2010, 12:41 PM

Originally Posted by General

OK.
Waiting for the other solutions.
I want to see another ideas

Thanks.

Ok, do a change of variable. $u = x - 1$ , so that as $x \to 1$ , $u \to 0$ . the limit becomes

$\lim_{u \to 0} \frac {(u + 1)^5 - 1}u$

now expand the top, cancel the $u$ 's, evaluate. do you like that one?

**General** · January 13th 2010, 12:42 PM

Originally Posted by Jhevon

Ok, do a change of variable. $u = x - 1$ , so that as $x \to 1$ , $u \to 0$ . the limit becomes

$\lim_{u \to 0} \frac {(u + 1)^5 - 1}u$

now expand the top, cancel the $u$ 's, evaluate

Its the same idea

Thank you.

**Krizalid** · January 13th 2010, 01:21 PM

Originally Posted by Jhevon

Ok, do a change of variable. $u = x - 1$ , so that as $x \to 1$ , $u \to 0$ . the limit becomes

$\lim_{u \to 0} \frac {(u + 1)^5 - 1}u$

now expand the top, cancel the $u$ 's, evaluate. do you like that one?

I wouldn't recommend that if having $\frac{x^{n}-1}{x-1},$ you wouldn't expand.

**Jhevon** · January 13th 2010, 01:33 PM

Originally Posted by Krizalid

I wouldn't recommend that if having $\frac{x^{n}-1}{x-1},$ you wouldn't expand.

Haha, point taken. for arbitrary n i would use one of the methods we have been restricted from using. here it's ok though

do you have another method for arbitrary (large) n values?

**CaptainBlack** · January 13th 2010, 02:11 PM

Originally Posted by General

How did you factor it ?
1 is zero of $x^5-1$
then (x-1) is a factor .. and you find $x^4+x^3+x^2+x+1$ by long division
which is not allowed to use.

You don't find this by long division you observe that $(x-1)(x^{n-1}+x^{n-2}+... +1)$ gives a telescoping sum. Or equivalently:

$x^n-1=[x^n+x^{n-1}+ ... +x] - [x^{n-1}+x^{n-2}+ ... +1]$

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

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

......... $= (x-1)[x^{n-1} + x^{n-2} + ... +1]$

CB

**Danny** · January 14th 2010, 07:21 AM

I think I can do it without what you said but using the ideas from calculus. Your limit represent the slope of the tangent of $y = x^5$ at $x = 1$ . Let

$\lim_{x \to 1} \frac{x^5-1}{x-1} = m$ .

The tangent line is $y-1 = m(x-1)$ or $y = mx -m +1$

The intersection of the tangent line and the quintic

$x^5 = mx - m + 1$ or $x^5 -mx + m -1 = 0 .\;\;(1.1)$

There should be two intersection points, at $x = 1$ and $x = a$ (some $a$ ) where the first will be a double root so eqn. (1.1) has the form

$ (x-1)^2(x-a)(x^2+px+q) = 0 \;\;(1.2) $ for some $p$ and $q$ .

Expanding and equating the coefficients of $(1.1)$ and $(1.2)$ gives

$(2.1)\;\;m + aq - 1 = 0,$
$(2.2)\;\;-m + ap - 2aq - q = 0,$
$ (2.3)\;\;aq - 2ap + 2q + a - p = 0, $
$ (2.4)\;\;ap + 2p - q - 2a - 1 =0, $
$(2.5)\;\;a-p+2 = 0.$

If we add/substract in the following order

$ (2.1) - (2.2)- 3 \times (2.3) - 5 \times (2.4) - 7 \times (2.5) = 0 $

gives

$2m - 10 = 0$ or $m = 5$ , our desired limit.

**I4talent** · January 14th 2010, 06:24 PM

Originally Posted by Jhevon

It's a known fact that $a^n - b^n = (a - b)(a^{n - 1} + a^{n - 2}b + a^{n - 3}b^2 + \cdots + ab^{n - 2} + b^{n - 1})$ when $n$ is an odd natural number

$a^n - b^n = (a-b)\left(\sum_{k=0}^{n-1}a^{n-k-1}b^{k}\right), \forall \ n\in \mathbb{N}$ .

It's $a^n + b^n = (a+b)\left(\sum_{k=0}^{n-1}(a^{n-k-1})(-b)^k\right)$ that only holds when n is an odd positive integer.

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