SOLUTION: Limx->1 {{{(x^8-1)/(x^7-1)}}} (Limit as x approaches 1)

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Question 921458: Limx->1 %28x%5E8-1%29%2F%28x%5E7-1%29
(Limit as x approaches 1)
Found 2 solutions by Alan3354, KMST:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Limx->1 %28x%5E8-1%29%2F%28x%5E7-1%29
(Limit as x approaches 1)
------------
Find the derivatives on the NUM and the DEN.
--> 8x%5E7%2F7x%5E6
Limit = 8/7

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There is probably a theorem that tells you what the limit should be, but who wants to memorize theorems?
We know that
x%5E8-1=%28x-1%29%28x%5E7%2Bx%5E6%2Bx%5E5%2Bx%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1%29
and
x%5E7-1=%28x-1%29%28x%5E6%2Bx%5E5%2Bx%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1%29 .
We can multiply and check, but we get the idea from what we remember.
We remember that
from geometric sequences sums,
or at least we can generalize from remembering from polynomials that
x%5E2-1=%28x-1%29%28x%2B1%29 and x%5E3-1=%28x-1%29%28x%5E2%2Bx%2B1%29 .
So, for x%3C%3E1 ,
,
and