1. ## Continuity

Hi everybody,

I must show that the following function accepts a limit in 1 :

$\displaystyle f(x)=\frac{x^{p+1}-(p+1)x+p}{(x-1)^2} (p\in \mathbb{N*})$

I know that i should factorize by (x-1) but how?

Thank's anyway.

2. Originally Posted by lehder
Hi everybody,

I must show that the following function accepts a limit in 1 :

$\displaystyle f(x)=\frac{x^{p+1}-(p+1)x+p}{(x-1)^2} (p\in \mathbb{N*})$

I know that i should factorize by (x-1) but how?

Thank's anyway.
Hi

$\displaystyle x^{p+1}-(p+1)x+p = x^{p+1}-1-(p+1)x+p+1$

$\displaystyle x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=0}^{p}x^k\right) -(p+1)(x-1)$

$\displaystyle x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=0}^{p}x^k -(p+1)\right)$

$\displaystyle x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=1}^{p}x^k -p\right)$

Let $\displaystyle g(x) = \sum_{k=1}^{p}x^k -p$

$\displaystyle g(1) = \sum_{k=1}^{p}1 -p = 0$

Therefore g(x) can be factored by (x-1)

3. Originally Posted by running-gag
hi

$\displaystyle x^{p+1}-(p+1)x+p = x^{p+1}-1-(p+1)x+p+1$

$\displaystyle x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=0}^{p}x^k\right) -(p+1)(x-1)$

ok thank you very much.

4. I have edited my post because you need to demonstrate that f can be factored by (x-1)²

5. Originally Posted by running-gag
i have edited my post because you need to demonstrate that f can be factored by (x-1)² (wink)
ok thank you again.