Limit question with Substituting

[Miike012]

Fine the limit of (x^1000 - 1)/(x - 1) as x approaches 1.

Solution:(x^1000 - 1)/(x - 1) = (x^999 + x^998 + ... + x + 1)(x - 1)/(x - 1)

= (x^999 + x^998 + ... + x + 1)

Substituting x = 1 I get...

Line (1):1 + 1₁ + 1₂ + ... + 1₉₉₉ = 1 + 999 = 1000

Is there any way to prove Line (1): I know it is obvious but I want to prove it mathematically and without obviously counting it on my fingers.

 

[Miike012]

I guess I could say 1 + 1(999) = 1000

 

[Miike012]

If anyone has a better way of solving it please let me know, I want to learn all I can.

 

[SammyS]

You could use synthetic division.

 

[Simon Bridge]

It's one of the surprising solutions that the limit as x --> 1 of (x^N -1)/(x-1) = N ... in the limit, the numerator is N times bigger than the denominator.

The question is to prove the expansion - you can demonstrate it simply enough by multiplying out the brackets so it is not clear what you mean when you want a non finger-counting method.

 

[Dick]

If anyone has a better way of solving it please let me know, I want to learn all I can.

You could use l'Hopital's theorem, if you have that.